Factor $5m^2 + 19m + 12$.A) $(3m + 4)(2m + 3)$ B) \$(5m + 2)(m + 6)$[/tex\] C) $(m + 4)(5m + 3)$ D) $(5m + 4)(m + 3)$

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Introduction

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Factoring quadratic expressions is a fundamental concept in algebra that involves expressing a quadratic expression as a product of two binomial expressions. In this article, we will focus on factoring the quadratic expression $5m^2 + 19m + 12$ and explore the different methods and techniques used to factorize it.

Understanding Quadratic Expressions

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A quadratic expression is a polynomial of degree two, which means it has a highest power of two. It can be written in the form $ax^2 + bx + c$, where $a$, $b$, and $c$ are constants, and $x$ is the variable. In our case, the quadratic expression is $5m^2 + 19m + 12$, where $a = 5$, $b = 19$, and $c = 12$.

Factoring Quadratic Expressions

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There are several methods used to factor quadratic expressions, including:

• Factoring by Grouping: This method involves grouping the terms of the quadratic expression into two groups and then factoring out the common factors from each group.
• Factoring by Difference of Squares: This method involves expressing the quadratic expression as a difference of squares, which can be factored into two binomial expressions.
• Factoring by Perfect Square Trinomials: This method involves expressing the quadratic expression as a perfect square trinomial, which can be factored into two binomial expressions.

Factoring the Quadratic Expression $5m^2 + 19m + 12$

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To factor the quadratic expression $5m^2 + 19m + 12$, we can use the method of factoring by grouping. This involves grouping the terms of the quadratic expression into two groups and then factoring out the common factors from each group.

Step 1: Group the Terms

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The first step is to group the terms of the quadratic expression into two groups. We can group the terms as follows:

$5m^2 + 19m + 12 = (5m^2 + 12m) + (7m + 12)$

Step 2: Factor Out the Common Factors

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The next step is to factor out the common factors from each group. We can factor out the common factors as follows:

$(5m^2 + 12m) + (7m + 12) = 5m(m + 2.4) + 7(m + 1.7)$

However, this is not the correct factorization. We need to find the correct factorization.

Step 3: Find the Correct Factorization

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To find the correct factorization, we need to find two numbers whose product is $5 \times 12 = 60$ and whose sum is $19$. These numbers are $15$ and $4$, since $15 \times 4 = 60$ and $15 + 4 = 19$.

Therefore, we can write the quadratic expression as:

$5m^2 + 19m + 12 = 5m^2 + 15m + 4m + 12$

Step 4: Factor Out the Common Factors

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The next step is to factor out the common factors from each group. We can factor out the common factors as follows:

$5m^2 + 15m + 4m + 12 = 5m(m + 3) + 4(m + 3)$

Step 5: Factor Out the Common Binomial Factor

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The final step is to factor out the common binomial factor from each group. We can factor out the common binomial factor as follows:

$5m(m + 3) + 4(m + 3) = (5m + 4)(m + 3)$

Therefore, the correct factorization of the quadratic expression $5m^2 + 19m + 12$ is $(5m + 4)(m + 3)$.

Conclusion

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In this article, we have explored the different methods and techniques used to factor quadratic expressions. We have also factored the quadratic expression $5m^2 + 19m + 12$ using the method of factoring by grouping. The correct factorization of the quadratic expression is $(5m + 4)(m + 3)$.

Discussion

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The discussion category for this article is mathematics. The article is relevant to the topic of factoring quadratic expressions and provides a step-by-step guide on how to factor the quadratic expression $5m^2 + 19m + 12$.

References

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• [1] "Factoring Quadratic Expressions" by Math Open Reference. Retrieved February 2023.
• [2] "Factoring Quadratic Expressions" by Purplemath. Retrieved February 2023.

Keywords

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• Factoring quadratic expressions
• Quadratic expressions
• Algebra
• Mathematics
• Factoring by grouping
• Factoring by difference of squares
• Factoring by perfect square trinomials
  

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Introduction

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In our previous article, we explored the different methods and techniques used to factor quadratic expressions. We also factored the quadratic expression $5m^2 + 19m + 12$ using the method of factoring by grouping. In this article, we will provide a Q&A guide to help you understand the concept of factoring quadratic expressions and how to apply it to different types of quadratic expressions.

Q&A

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Q: What is a quadratic expression?

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A: A quadratic expression is a polynomial of degree two, which means it has a highest power of two. It can be written in the form $ax^2 + bx + c$, where $a$, $b$, and $c$ are constants, and $x$ is the variable.

Q: What are the different methods used to factor quadratic expressions?

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A: There are several methods used to factor quadratic expressions, including:

• Factoring by Grouping: This method involves grouping the terms of the quadratic expression into two groups and then factoring out the common factors from each group.
• Factoring by Difference of Squares: This method involves expressing the quadratic expression as a difference of squares, which can be factored into two binomial expressions.
• Factoring by Perfect Square Trinomials: This method involves expressing the quadratic expression as a perfect square trinomial, which can be factored into two binomial expressions.

Q: How do I factor a quadratic expression using the method of factoring by grouping?

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A: To factor a quadratic expression using the method of factoring by grouping, follow these steps:

1. Group the terms of the quadratic expression into two groups.
2. Factor out the common factors from each group.
3. Factor out the common binomial factor from each group.

Q: How do I factor a quadratic expression using the method of factoring by difference of squares?

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A: To factor a quadratic expression using the method of factoring by difference of squares, follow these steps:

1. Express the quadratic expression as a difference of squares.
2. Factor the difference of squares into two binomial expressions.

Q: How do I factor a quadratic expression using the method of factoring by perfect square trinomials?

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A: To factor a quadratic expression using the method of factoring by perfect square trinomials, follow these steps:

1. Express the quadratic expression as a perfect square trinomial.
2. Factor the perfect square trinomial into two binomial expressions.

Q: What are some common mistakes to avoid when factoring quadratic expressions?

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A: Some common mistakes to avoid when factoring quadratic expressions include:

• Not grouping the terms correctly.
• Not factoring out the common factors correctly.
• Not factoring out the common binomial factor correctly.

Q: How do I check if my factorization is correct?

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A: To check if your factorization is correct, follow these steps:

1. Multiply the two binomial expressions together.
2. Simplify the result.
3. Check if the result is equal to the original quadratic expression.

Conclusion

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In this article, we have provided a Q&A guide to help you understand the concept of factoring quadratic expressions and how to apply it to different types of quadratic expressions. We have also discussed some common mistakes to avoid when factoring quadratic expressions and how to check if your factorization is correct.

Discussion

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The discussion category for this article is mathematics. The article is relevant to the topic of factoring quadratic expressions and provides a Q&A guide to help you understand the concept.

References

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• [1] "Factoring Quadratic Expressions" by Math Open Reference. Retrieved February 2023.
• [2] "Factoring Quadratic Expressions" by Purplemath. Retrieved February 2023.

Keywords

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• Factoring quadratic expressions
• Quadratic expressions
• Algebra
• Mathematics
• Factoring by grouping
• Factoring by difference of squares
• Factoring by perfect square trinomials
• Q&A guide
• Common mistakes to avoid
• Checking factorization