What Is The Product Of $a+3$ And $-2a^2+15a+6b^2$?A. $-2a^3+9a^2+45a+24b^2$B. $-2a^3+21a^2+45a+24b^2$C. $-2a^3+9a^2+45a+6ab^2+18b^2$D. $-2a^3+21a^2+45a+6ab^2+18b^2$

What is the Product of $a+3$ and $-2a^2+15a+6b^2$?

Understanding the Problem

The problem requires us to find the product of two algebraic expressions: $a+3$ and $-2a^2+15a+6b^2$. To solve this problem, we need to apply the distributive property of multiplication over addition, which states that the product of a number and a sum is equal to the sum of the products of the number and each term in the sum.

Step 1: Apply the Distributive Property

To find the product of $a+3$ and $-2a^2+15a+6b^2$, we need to multiply each term in the first expression by each term in the second expression. This can be done using the distributive property, which states that:

$(a+3) \cdot (-2a^2+15a+6b^2) = a \cdot (-2a^2+15a+6b^2) + 3 \cdot (-2a^2+15a+6b^2)$

Step 2: Multiply Each Term

Now, we need to multiply each term in the first expression by each term in the second expression.

$a \cdot (-2a^2+15a+6b^2) = -2a^3+15a^2+6ab^2$

$3 \cdot (-2a^2+15a+6b^2) = -6a^2+45a+18b^2$

Step 3: Combine Like Terms

Now, we need to combine like terms to simplify the expression.

$-2a^3+15a^2+6ab^2-6a^2+45a+18b^2$

Combining like terms, we get:

$-2a^3+9a^2+6ab^2+45a+18b^2$

Step 4: Simplify the Expression

The expression $-2a^3+9a^2+6ab^2+45a+18b^2$ is the product of $a+3$ and $-2a^2+15a+6b^2$. However, we need to check if this expression matches any of the answer choices.

Answer Choice Comparison

Let's compare the expression $-2a^3+9a^2+6ab^2+45a+18b^2$ with the answer choices.

A. $-2a^3+9a^2+45a+24b^2$

B. $-2a^3+21a^2+45a+24b^2$

C. $-2a^3+9a^2+45a+6ab^2+18b^2$

D. $-2a^3+21a^2+45a+6ab^2+18b^2$

The expression $-2a^3+9a^2+6ab^2+45a+18b^2$ matches answer choice C.

Conclusion

The product of $a+3$ and $-2a^2+15a+6b^2$ is $-2a^3+9a^2+6ab^2+45a+18b^2$, which matches answer choice C.

Key Takeaways

• The distributive property of multiplication over addition states that the product of a number and a sum is equal to the sum of the products of the number and each term in the sum.
• To find the product of two algebraic expressions, we need to apply the distributive property and multiply each term in the first expression by each term in the second expression.
• Combining like terms is an important step in simplifying algebraic expressions.

Final Answer

The final answer is C.
What is the Product of $a+3$ and $-2a^2+15a+6b^2$? - Q&A

Understanding the Problem

The problem requires us to find the product of two algebraic expressions: $a+3$ and $-2a^2+15a+6b^2$. To solve this problem, we need to apply the distributive property of multiplication over addition, which states that the product of a number and a sum is equal to the sum of the products of the number and each term in the sum.

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

A: The distributive property of multiplication over addition states that the product of a number and a sum is equal to the sum of the products of the number and each term in the sum.

Q: How do we apply the distributive property to find the product of two algebraic expressions?

A: To find the product of two algebraic expressions, we need to multiply each term in the first expression by each term in the second expression. This can be done using the distributive property, which states that:

$(a+3) \cdot (-2a^2+15a+6b^2) = a \cdot (-2a^2+15a+6b^2) + 3 \cdot (-2a^2+15a+6b^2)$

Q: What is the product of $a$ and $-2a^2+15a+6b^2$?

A: The product of $a$ and $-2a^2+15a+6b^2$ is $-2a^3+15a^2+6ab^2$.

Q: What is the product of $3$ and $-2a^2+15a+6b^2$?

A: The product of $3$ and $-2a^2+15a+6b^2$ is $-6a^2+45a+18b^2$.

Q: How do we combine like terms to simplify the expression?

A: To combine like terms, we need to add or subtract the coefficients of the same variables. In this case, we have:

$-2a^3+15a^2+6ab^2-6a^2+45a+18b^2$

Combining like terms, we get:

$-2a^3+9a^2+6ab^2+45a+18b^2$

Q: What is the final answer?

A: The final answer is $-2a^3+9a^2+6ab^2+45a+18b^2$.

Key Takeaways

• The distributive property of multiplication over addition states that the product of a number and a sum is equal to the sum of the products of the number and each term in the sum.
• To find the product of two algebraic expressions, we need to apply the distributive property and multiply each term in the first expression by each term in the second expression.
• Combining like terms is an important step in simplifying algebraic expressions.

Frequently Asked Questions

• Q: What is the distributive property of multiplication over addition? A: The distributive property of multiplication over addition states that the product of a number and a sum is equal to the sum of the products of the number and each term in the sum.
• Q: How do we apply the distributive property to find the product of two algebraic expressions? A: To find the product of two algebraic expressions, we need to multiply each term in the first expression by each term in the second expression.
• Q: What is the product of $a$ and $-2a^2+15a+6b^2$? A: The product of $a$ and $-2a^2+15a+6b^2$ is $-2a^3+15a^2+6ab^2$.
• Q: What is the product of $3$ and $-2a^2+15a+6b^2$? A: The product of $3$ and $-2a^2+15a+6b^2$ is $-6a^2+45a+18b^2$.

Conclusion

The product of $a+3$ and $-2a^2+15a+6b^2$ is $-2a^3+9a^2+6ab^2+45a+18b^2$. This can be found by applying the distributive property of multiplication over addition and combining like terms.