Clara found the product of \(3 - 6 y^2\) and \(y^2 + 2\). Her work is shown below.

[tex]\[
\begin{aligned}
(3 & - 6 y^2)(y^2 + 2) = 3(y^2) + (-6 y^2)(2) \\
& = 3 y^2 - 12 y^2 \\
& = -9 y^2
\end{aligned}
\][/tex]

Is the student's work correct?

A. No, she did not multiply \(-6 y^2\) by 2 correctly.
B. No, she did not add \(3 y^2\) and \(-12 y^2\) correctly.
C. No, she did not use the distributive property correctly.
D. Yes, she multiplied the binomials correctly.



Answer :

Clara’s task is to find the product of \((3 - 6y^2)\) and \((y^2 + 2)\). Let’s work through this step-by-step using the distributive property to make sure all steps are handled properly.

Step 1: Apply the distributive property
[tex]\[ (3 - 6y^2)(y^2 + 2) = 3(y^2 + 2) - 6y^2(y^2 + 2) \][/tex]

Step 2: Distribute each term
[tex]\[ = 3 \cdot y^2 + 3 \cdot 2 - 6y^2 \cdot y^2 - 6y^2 \cdot 2 \][/tex]

Step 3: Perform the multiplications
[tex]\[ = 3y^2 + 6 - 6y^4 - 12y^2 \][/tex]

Step 4: Combine like terms
[tex]\[ = -6y^4 + 3y^2 - 12y^2 + 6 \][/tex]
[tex]\[ = -6y^4 - 9y^2 + 6 \][/tex]

Now, let’s compare this result to Clara’s work:
Clara’s work:
[tex]\[ (3 - 6y^2)(y^2 + 2) = 3(y^2) + (-6y^2)(2) = 3y^2 - 12y^2 = -9y^2 \][/tex]

Clearly, Clara did not perform all the necessary steps, specifically:
1. She did not multiply 3 by 2.
2. She did not include all necessary terms resulting from the distribution.

Therefore, the correct conclusion is:
No, she did not use the distributive property correctly.

The correct product is:
[tex]\[ -6y^4 - 9y^2 + 6 \][/tex]