Let's go through the problem step by step and find the product.
We are given the expression:
[tex]\[ 2y^2(3x + 5z) \][/tex]
To solve this, we need to distribute [tex]\(2y^2\)[/tex] to each term inside the parentheses.
1. Distribute [tex]\(2y^2\)[/tex] to [tex]\(3x\)[/tex]:
[tex]\[
2y^2 \cdot 3x = 6xy^2
\][/tex]
2. Distribute [tex]\(2y^2\)[/tex] to [tex]\(5z\)[/tex]:
[tex]\[
2y^2 \cdot 5z = 10y^2z
\][/tex]
Putting these two results together, we get:
[tex]\[
2y^2(3x + 5z) = 6xy^2 + 10y^2z
\][/tex]
Now, let's compare this result with the given possible answers:
1. [tex]\(5xy^2 + 7y^2z\)[/tex]
2. [tex]\(6xy^2 + 10y^2z^2\)[/tex]
3. [tex]\(6xy^2 + 10y^2z\)[/tex]
From our calculations, we see that the product [tex]\(2y^2(3x + 5z)\)[/tex] simplifies to [tex]\(6xy^2 + 10y^2z\)[/tex].
Therefore, the correct answer is:
[tex]\[ 6xy^2 + 10y^2z \][/tex]
