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]
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]
