Algebra worksheet featuring polynomial multiplication exercises.

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The image shows a math worksheet titled "Multiplying Polynomials". It contains 12 problems. I will solve each one step by step.

Problem 1:
$(x + 2)(x - 5)$
Using the FOIL method (First, Outer, Inner, Last):
- First: $x \cdot x = x^2$
- Outer: $x \cdot (-5) = -5x$
- Inner: $2 \cdot x = 2x$
- Last: $2 \cdot (-5) = -10$

Combine like terms: $x^2 - 5x + 2x - 10 = x^2 - 3x - 10$

Problem 2:
$(x + 3)(x + 4)$
Using FOIL:
- First: $x \cdot x = x^2$
- Outer: $x \cdot 4 = 4x$
- Inner: $3 \cdot x = 3x$
- Last: $3 \cdot 4 = 12$

Combine like terms: $x^2 + 4x + 3x + 12 = x^2 + 7x + 12$

Problem 3:
$(x - 6)(x - 3)$
Using FOIL:
- First: $x \cdot x = x^2$
- Outer: $x \cdot (-3) = -3x$
- Inner: $-6 \cdot x = -6x$
- Last: $-6 \cdot (-3) = 18$

Combine like terms: $x^2 - 3x - 6x + 18 = x^2 - 9x + 18$

Problem 4:
$(x + 8)(x + 1)$
Using FOIL:
- First: $x \cdot x = x^2$
- Outer: $x \cdot 1 = x$
- Inner: $8 \cdot x = 8x$
- Last: $8 \cdot 1 = 8$

Combine like terms: $x^2 + x + 8x + 8 = x^2 + 9x + 8$

Problem 5:
$(x + 5)(x + 5)$
Using FOIL:
- First: $x \cdot x = x^2$
- Outer: $x \cdot 5 = 5x$
- Inner: $5 \cdot x = 5x$
- Last: $5 \cdot 5 = 25$

Combine like terms: $x^2 + 5x + 5x + 25 = x^2 + 10x + 25$

Problem 6:
$(x + 2)(x^2 + 3x + 4)$
Distribute each term in the first polynomial to every term in the second:
- $x(x^2 + 3x + 4) = x^3 + 3x^2 + 4x$
- $2(x^2 + 3x + 4) = 2x^2 + 6x + 8$

Combine like terms: $x^3 + 3x^2 + 2x^2 + 4x + 6x + 8 = x^3 + 5x^2 + 10x + 8$

Problem 7:
$(x - 4)(x^2 - 2x + 3)$
Distribute:
- $x(x^2 - 2x + 3) = x^3 - 2x^2 + 3x$
- $-4(x^2 - 2x + 3) = -4x^2 + 8x - 12$

Combine like terms: $x^3 - 2x^2 - 4x^2 + 3x + 8x - 12 = x^3 - 6x^2 + 11x - 12$

Problem 8:
$(x + 1)(x^2 + 4x - 5)$
Distribute:
- $x(x^2 + 4x - 5) = x^3 + 4x^2 - 5x$
- $1(x^2 + 4x - 5) = x^2 + 4x - 5$

Combine like terms: $x^3 + 4x^2 + x^2 - 5x + 4x - 5 = x^3 + 5x^2 - x - 5$

Problem 9:
$(x + 3)(x^2 - 5x + 6)$
Distribute:
- $x(x^2 - 5x + 6) = x^3 - 5x^2 + 6x$
- $3(x^2 - 5x + 6) = 3x^2 - 15x + 18$

Combine like terms: $x^3 - 5x^2 + 3x^2 + 6x - 15x + 18 = x^3 - 2x^2 - 9x + 18$

Problem 10:
$(x - 2)(x^2 + 3x - 7)$
Distribute:
- $x(x^2 + 3x - 7) = x^3 + 3x^2 - 7x$
- $-2(x^2 + 3x - 7) = -2x^2 - 6x + 14$

Combine like terms: $x^3 + 3x^2 - 2x^2 - 7x - 6x + 14 = x^3 + x^2 - 13x + 14$

Problem 11:
$(x - 5)(x^2 - 4x + 9)$
Distribute:
- $x(x^2 - 4x + 9) = x^3 - 4x^2 + 9x$
- $-5(x^2 - 4x + 9) = -5x^2 + 20x - 45$

Combine like terms: $x^3 - 4x^2 - 5x^2 + 9x + 20x - 45 = x^3 - 9x^2 + 29x - 45$

Problem 12:
$(x + 4)(x^2 - 6x + 8)$
Distribute:
- $x(x^2 - 6x + 8) = x^3 - 6x^2 + 8x$
- $4(x^2 - 6x + 8) = 4x^2 - 24x + 32$

Combine like terms: $x^3 - 6x^2 + 4x^2 + 8x - 24x + 32 = x^3 - 2x^2 - 16x + 32$

Final Answer:
1. $x^2 - 3x - 10$
2. $x^2 + 7x + 12$
3. $x^2 - 9x + 18$
4. $x^2 + 9x + 8$
5. $x^2 + 10x + 25$
6. $x^3 + 5x^2 + 10x + 8$
7. $x^3 - 6x^2 + 11x - 12$
8. $x^3 + 5x^2 - x - 5$
9. $x^3 - 2x^2 - 9x + 18$
10. $x^3 + x^2 - 13x + 14$
11. $x^3 - 9x^2 + 29x - 45$
12. $x^3 - 2x^2 - 16x + 32$