What Is The Product?\[$(4s + 2)(5s^2 + 10s + 3)\$\]A. \[$20s^2 + 20s + 6\$\] B. \[$20s^3 + 40s^2 + 12s\$\] C. \[$20s^3 + 10s^2 + 32s + 6\$\] D. \[$20s^3 + 50s^2 + 32s + 6\$\]

Understanding the Problem

The given problem involves finding the product of two polynomial expressions. To solve this, we need to apply the distributive property of multiplication over addition, which states that the product of a sum is equal to the sum of the products.

The Given Expression

The given expression is $(4s + 2)(5s^2 + 10s + 3)$. To find the product, we need to multiply each term in the first expression by each term in the second expression.

Multiplying the Expressions

To multiply the expressions, we will use the distributive property. We will multiply each term in the first expression by each term in the second expression and then combine like terms.

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 $4s$. We will multiply this by each term in the second expression:

• $4s \times 5s^2 = 20s^3$
• $4s \times 10s = 40s^2$
• $4s \times 3 = 12s$

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 $2$. We will multiply this by each term in the second expression:

• $2 \times 5s^2 = 10s^2$
• $2 \times 10s = 20s$
• $2 \times 3 = 6$

Step 3: Combine Like Terms

Now, we will combine like terms:

• $20s^3$ (no like terms)
• $40s^2 + 10s^2 = 50s^2$
• $12s + 20s = 32s$
• $6$ (no like terms)

The Final Product

The final product is $20s^3 + 50s^2 + 32s + 6$.

Comparing the Final Product with the Options

Now, we will compare the final product with the options:

• A. $20s^2 + 20s + 6$ ( incorrect, missing $s^3$ term)
• B. $20s^3 + 40s^2 + 12s$ ( incorrect, missing $50s^2$ term)
• C. $20s^3 + 10s^2 + 32s + 6$ ( incorrect, missing $50s^2$ term)
• D. $20s^3 + 50s^2 + 32s + 6$ ( correct)

The final answer is D.

Understanding the Problem

The given problem involves finding the product of two polynomial expressions. To solve this, we need to apply the distributive property of multiplication over addition, which states that the product of a sum is equal to the sum of the products.

Q&A

Q: What is the distributive property of multiplication over addition?

A: The distributive property of multiplication over addition states that the product of a sum is equal to the sum of the products. This means that we can multiply each term in one expression by each term in the other expression and then combine like terms.

Q: How do I multiply two polynomial expressions?

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

Q: What is the product of the given expression $(4s + 2)(5s^2 + 10s + 3)$?

A: The product of the given expression is $20s^3 + 50s^2 + 32s + 6$.

Q: Why is the product of the given expression $20s^3 + 50s^2 + 32s + 6$?

A: The product of the given expression is $20s^3 + 50s^2 + 32s + 6$ because we multiplied each term in the first expression by each term in the second expression and then combined like terms.

Q: How do I compare the final product with the options?

A: To compare the final product with the options, we need to look at each option and see if it matches the final product. If it does, then that option is correct.

Q: What is the correct option?

A: The correct option is D. $20s^3 + 50s^2 + 32s + 6$.

Common Mistakes to Avoid

Mistake 1: Not Applying the Distributive Property

• Not applying the distributive property can lead to incorrect results.
• Make sure to multiply each term in one expression by each term in the other expression.

Mistake 2: Not Combining Like Terms

• Not combining like terms can lead to incorrect results.
• Make sure to combine like terms after multiplying the expressions.

Mistake 3: Not Comparing the Final Product with the Options

• Not comparing the final product with the options can lead to incorrect results.
• Make sure to compare the final product with the options to see which one is correct.

Conclusion

Finding the product of two polynomial expressions involves applying the distributive property of multiplication over addition. We need to multiply each term in one expression by each term in the other expression and then combine like terms. By following these steps, we can find the correct product of the given expression.

Final Answer

The final answer is D. $20s^3 + 50s^2 + 32s + 6$.