What Is The Product Of 6 X − Y 6x - Y 6 X − Y And 2 X − Y + 2 2x - Y + 2 2 X − Y + 2 ?A. 8 X 2 − 4 X Y + 12 X + Y 2 − 2 Y 8x^2 - 4xy + 12x + Y^2 - 2y 8 X 2 − 4 X Y + 12 X + Y 2 − 2 Y B. 12 X 2 − 8 X Y + 12 X + Y 2 − 2 Y 12x^2 - 8xy + 12x + Y^2 - 2y 12 X 2 − 8 X Y + 12 X + Y 2 − 2 Y C. 8 X 2 + 4 X Y + 4 X + Y 2 − 2 Y 8x^2 + 4xy + 4x + Y^2 - 2y 8 X 2 + 4 X Y + 4 X + Y 2 − 2 Y D. 12 X 2 + 8 X Y + 4 X + Y 2 + 2 Y 12x^2 + 8xy + 4x + Y^2 + 2y 12 X 2 + 8 X Y + 4 X + Y 2 + 2 Y

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Understanding the Problem

To find the product of two expressions, we need to multiply each term in the first expression by each term in the second expression and then combine like terms. In this case, we are given the expressions $6x - y$ and $2x - y + 2$. Our goal is to find the product of these two expressions.

Multiplying the Expressions

To multiply the expressions, we will use the distributive property, which states that for any real numbers $a$, $b$, and $c$, $a(b + c) = ab + ac$. We will apply this property to each term in the first expression and multiply it by each term in the second expression.

Step 1: Multiply the first term in the first expression by each term in the second expression

The first term in the first expression is $6x$. We will multiply this term by each term in the second expression, which are $2x$, $-y$, and $2$.

• $6x \cdot 2x = 12x^2$
• $6x \cdot (-y) = -6xy$
• $6x \cdot 2 = 12x$

Step 2: Multiply the second term in the first expression by each term in the second expression

The second term in the first expression is $-y$. We will multiply this term by each term in the second expression, which are $2x$, $-y$, and $2$.

• $(-y) \cdot 2x = -2xy$
• $(-y) \cdot (-y) = y^2$
• $(-y) \cdot 2 = -2y$

Combining Like Terms

Now that we have multiplied each term in the first expression by each term in the second expression, we need to combine like terms. Like terms are terms that have the same variable(s) raised to the same power.

• The terms $12x^2$ and $-2xy$ are like terms, but they have different variables, so we cannot combine them.
• The terms $-6xy$ and $-2xy$ are like terms, so we can combine them to get $-8xy$.
• The terms $12x$ and $-2y$ are like terms, but they have different variables, so we cannot combine them.
• The term $y^2$ is a like term with itself, so we can leave it as is.

Writing the Final Answer

After combining like terms, we get the following expression:

$12x^2 - 8xy + 12x + y^2 - 2y$

This is the product of the expressions $6x - y$ and $2x - y + 2$.

Conclusion

In this article, we used the distributive property to multiply two expressions and then combined like terms to get the final answer. We also discussed the importance of understanding the problem and following the correct steps to solve it.

Final Answer

The final answer is: $12x^2 - 8xy + 12x + y^2 - 2y$

This answer matches option B in the given choices.

Understanding the Basics of Multiplying Expressions

Multiplying expressions is a fundamental concept in algebra that involves combining like terms and applying the distributive property. In this article, we will answer some frequently asked questions about multiplying expressions.

Q: What is the distributive property?

A: The distributive property is a mathematical concept that states that for any real numbers $a$, $b$, and $c$, $a(b + c) = ab + ac$. This means that we can multiply a single term by a sum of terms by multiplying the single term by each term in the sum and then combining the results.

Q: How do I multiply two expressions?

A: To multiply two expressions, we need to apply the distributive property to each term in the first expression and multiply it by each term in the second expression. We then combine like terms to get the final answer.

Q: What are like terms?

A: Like terms are terms that have the same variable(s) raised to the same power. For example, $2x$ and $-3x$ are like terms because they both have the variable $x$ raised to the power of 1. We can combine like terms by adding or subtracting their coefficients.

Q: How do I combine like terms?

A: To combine like terms, we need to add or subtract their coefficients. For example, if we have the terms $2x$ and $-3x$, we can combine them by adding their coefficients to get $-x$.

Q: What is the product of $6x - y$ and $2x - y + 2$?

A: The product of $6x - y$ and $2x - y + 2$ is $12x^2 - 8xy + 12x + y^2 - 2y$. This is obtained by applying the distributive property and combining like terms.

Q: How do I know which terms are like terms?

A: To determine which terms are like terms, we need to look at the variables and their exponents. If the variables and exponents are the same, then the terms are like terms.

Q: Can I multiply expressions with variables that have different exponents?

A: Yes, you can multiply expressions with variables that have different exponents. However, you need to be careful when combining like terms, as the exponents will be different.

Q: What is the final answer to the problem of multiplying $6x - y$ and $2x - y + 2$?

A: The final answer to the problem of multiplying $6x - y$ and $2x - y + 2$ is $12x^2 - 8xy + 12x + y^2 - 2y$.

Conclusion

In this article, we have answered some frequently asked questions about multiplying expressions. We have discussed the distributive property, like terms, and combining like terms. We have also provided examples and explanations to help you understand the concepts better.

Final Answer

The final answer is: $12x^2 - 8xy + 12x + y^2 - 2y$

This answer matches option B in the given choices.

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If you want to learn more about multiplying expressions, here are some additional resources that you can use:

• Khan Academy: Multiplying Expressions
• Mathway: Multiplying Expressions
• Wolfram Alpha: Multiplying Expressions

These resources provide step-by-step explanations and examples to help you understand the concepts better.

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Here are some practice problems that you can use to test your understanding of multiplying expressions:

1. Multiply $3x + 2y$ and $2x + 3y$.
2. Multiply $4x - 2y$ and $3x + 2y$.
3. Multiply $2x + 3y$ and $4x - 2y$.

Solve these problems and check your answers with the solutions provided below.

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1. $6x^2 + 6xy + 4xy + 6y^2 = 6x^2 + 10xy + 6y^2$
2. $12x^2 - 4xy - 6xy + 4y^2 = 12x^2 - 10xy + 4y^2$
3. $8x^2 + 6xy - 8xy - 6y^2 = 8x^2 - 2xy - 6y^2$

I hope these resources and practice problems help you understand multiplying expressions better.