Find The Product. { (5j + 1)(4 + J)$}$A. ${ 5j^2 + 21j - 4\$} B. ${ 5j^2 + 4\$} C. ${ 5j^2 + 21j + 4\$} D. ${ 5j^2 + 20j + 4\$}

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Understanding the Problem

To find the product of two binomials, we can use the distributive property, which states that for any real numbers $a$, $b$, and $c$, $a(b + c) = ab + ac$. This property allows us to multiply each term in the first binomial by each term in the second binomial.

Applying the Distributive Property

We will apply the distributive property to find the product of $(5j + 1)$ and $(4 + j)$. To do this, we will multiply each term in the first binomial by each term in the second binomial.

Multiplying the Terms

First, we will multiply $5j$ by $4$:

$$5j \cdot 4 = 20j$$

Next, we will multiply $5j$ by $j$:

$$5j \cdot j = 5j^2$$

Then, we will multiply $1$ by $4$:

$$1 \cdot 4 = 4$$

Finally, we will multiply $1$ by $j$:

$$1 \cdot j = j$$

Combining the Terms

Now that we have multiplied each term in the first binomial by each term in the second binomial, we can combine the terms to find the product.

$$20j + 5j^2 + 4 + j$$

Simplifying the Expression

We can simplify the expression by combining like terms. The like terms in this expression are the terms with the same variable, which are $20j$ and $j$.

$$20j + j = 21j$$

So, the simplified expression is:

$$5j^2 + 21j + 4$$

Conclusion

The product of $(5j + 1)$ and $(4 + j)$ is $5j^2 + 21j + 4$. This is the correct answer.

Comparison with the Options

Let's compare our answer with the options provided:

• Option A: $5j^2 + 21j - 4$
• Option B: $5j^2 + 4$
• Option C: $5j^2 + 21j + 4$
• Option D: $5j^2 + 20j + 4$

Our answer, $5j^2 + 21j + 4$, matches option C.

Final Answer

The final answer is option C: $5j^2 + 21j + 4$.

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Understanding the Problem

To find the product of two binomials, we can use the distributive property, which states that for any real numbers $a$, $b$, and $c$, $a(b + c) = ab + ac$. This property allows us to multiply each term in the first binomial by each term in the second binomial.

Q&A

Q: What is the distributive property?

A: The distributive property is a mathematical property that allows us to multiply each term in the first binomial by each term in the second binomial.

Q: How do I apply the distributive property to find the product of two binomials?

A: To apply the distributive property, you need to multiply each term in the first binomial by each term in the second binomial. This will give you the product of the two binomials.

Q: What are like terms?

A: Like terms are terms with the same variable. For example, $20j$ and $j$ are like terms because they both have the variable $j$.

Q: How do I simplify an expression by combining like terms?

A: To simplify an expression by combining like terms, you need to add or subtract the coefficients of the like terms. For example, $20j + j = 21j$.

Q: What is the product of $(5j + 1)$ and $(4 + j)$?

A: The product of $(5j + 1)$ and $(4 + j)$ is $5j^2 + 21j + 4$.

Q: Which option is correct?

A: The correct option is option C: $5j^2 + 21j + 4$.

Common Mistakes

• Not applying the distributive property correctly
• Not combining like terms correctly
• Not simplifying the expression correctly

Tips and Tricks

• Make sure to apply the distributive property correctly
• Combine like terms carefully
• Simplify the expression by combining like terms

Conclusion

Finding the product of two binomials using the distributive property can be a challenging task, but with practice and patience, you can master it. Remember to apply the distributive property correctly, combine like terms carefully, and simplify the expression by combining like terms.

Final Answer

The final answer is option C: $5j^2 + 21j + 4$.