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The problem in the image asks to match the products on the left-hand side with their respective results on the right-hand side. Let's go step by step and solve each multiplication expression from the left-hand side to find the corresponding result.

### a. \((9x^4 + y^2)(x^3y^2z)\)
Distribute the terms:
\[
9x^4(x^3y^2z) + y^2(x^3y^2z) = 9x^7y^2z + x^3y^4z
\]
So, this matches with:
\[
9x^7y^2z + x^3y^4z
\]

This corresponds to option **c** on the right-hand side.

### b. \((x^2)(3x^3y^4 + z^2)\)
Distribute the terms:
\[
x^2(3x^3y^4) + x^2(z^2) = 3x^5y^4 + x^2z^2
\]
This matches with:
\[
3x^5y^4 + x^2z^2
\]
So, this corresponds to option **d**.

### c. \((-3y^2)(3x^3y^4 + z^2)\)
Distribute the terms:
\[
-3y^2(3x^3y^4) + (-3y^2)(z^2) = -9x^3y^6 - 3y^2z^2
\]
This matches with:
\[
-9x^3y^6 - 3y^2z^2
\]
So, this corresponds to option **a**.

### d. \((2x^4y)(2x^4 - y^2z^2)\)
Distribute the terms:
\[
2x^4y(2x^4) + 2x^4y(-y^2z^2) = 4x^8y - 2x^4y^3z^2
\]
This matches with:
\[
4x^8y - 2x^4y^3z^2
\]
So, this corresponds to option **f**.

### e. \((-3x^4 + y)(-2xy^3)\)
Distribute the terms:
\[
-3x^4(-2xy^3) + y(-2xy^3) = 6x^5y^3 - 2xy^4
\]
This matches with:
\[
6x^5y^3 - 2xy^4
\]
So, this corresponds to option **b**.

### f. \((-4x^3 - y)(4xy^2)\)
Distribute the terms:
\[
-4x^3(4xy^2) + (-y)(4xy^2) = -16x^4y^2 - 4xy^3
\]
This matches with:
\[
-16x^4y^2 - 4xy^3
\]
So, this corresponds to option **e**.

### Final Matching:
- a → c
- b → d
- c → a
- d → f
- e → b
- f → e

Would you like more details on any of these steps or further clarifications?

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Here are 5 related questions for practice:
1. How would you simplify the expression \((x^2 + y)(3x^3 - y^2)\)?
2. What is the result of multiplying \(2x^4\) by \((x^3 + 3xy)\)?
3. How can you factor the polynomial \(6x^5y^3 - 2xy^4\)?
4. What is the expanded form of \((x^2y + z)(x^3z - y^2)\)?
5. How would you factor \(-16x^4y^2 - 4xy^3\)?

**Tip:** When distributing terms, remember to apply the distributive property carefully to each term inside the parentheses to avoid mistakes.

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This problem involves matching products of polynomials with their results. It requires knowledge of polynomial multiplication and the distributive property. Suitable for students in Grades 9-12, this problem helps strengthen algebraic skills.

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