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?

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.