To find the product given expressions and then compare them with specific terms, let’s break down the solution step by step.
Step 1: Define and simplify the given expressions.
We are given two expressions:
1. [tex]\(2y^2(3x + 5z)\)[/tex]
2. [tex]\(5xy^2 + 7y^2z\)[/tex]
Let's simplify each expression individually.
Simplifying [tex]\(2y^2(3x + 5z)\)[/tex]:
[tex]\[
2y^2(3x + 5z) = 2y^2 \cdot 3x + 2y^2 \cdot 5z = 6xy^2 + 10y^2z
\][/tex]
Thus, the simplified form of [tex]\(2y^2(3x + 5z)\)[/tex] is:
[tex]\[
6xy^2 + 10y^2z
\][/tex]
Simplifying [tex]\(5xy^2 + 7y^2z\)[/tex]:
The expression [tex]\(5xy^2 + 7y^2z\)[/tex] is already in its simplest form.
Step 2: Compare the given terms with the simplified expressions.
We need to establish whether the given terms match with our simplified expressions:
1. [tex]\(6xy^2 + 10y^2z^2\)[/tex]
2. [tex]\(6xy^2 + 10y^2z\)[/tex]
The expressions we have calculated are:
1. [tex]\(6xy^2 + 10y^2z\)[/tex]
2. [tex]\(5xy^2 + 7y^2z\)[/tex]
Comparing these with the given terms, it is clear that:
- [tex]\(6xy^2 + 10y^2z\)[/tex] is already found exactly in our simplifications.
- [tex]\(5xy^2 + 7y^2z\)[/tex] matches with one of our originally given expressions.
Step 3: Finalize the product.
Thus, the step-by-step evaluation indicates that the relevant products involving the given terms and expressions are consistent with:
- [tex]\(6xy^2 + 10y^2z^2\)[/tex]
- [tex]\(6xy^2 + 10y^2z\)[/tex]
Therefore, the answer to the product is:
[tex]\[
(6xy^2 + 10y^2z^2, 6xy^2 + 10y^2z)
\][/tex]
Additionally validated terms:
[tex]\[
6xy^2 + 10y^2z
\][/tex]
[tex]\[
5xy^2 + 7y^2z
\][/tex]
