Multiplying Polynomials Coloring Activity worksheet featuring ten algebraic expressions to simplify.

Here are the simplified expressions for each problem. I have worked through them step-by-step to ensure accuracy.

1. $(a + 4)(a + 7)$
* Multiply $a \cdot a = a^2$
* Multiply $a \cdot 7 = 7a$
* Multiply $4 \cdot a = 4a$
* Multiply $4 \cdot 7 = 28$
* Combine like terms: $7a + 4a = 11a$
* Result: $a^2 + 11a + 28$

2. $(a - 7)(a - 3)$
* Multiply $a \cdot a = a^2$
* Multiply $a \cdot -3 = -3a$
* Multiply $-7 \cdot a = -7a$
* Multiply $-7 \cdot -3 = 21$ (negative times negative is positive)
* Combine like terms: $-3a - 7a = -10a$
* Result: $a^2 - 10a + 21$

3. $(a - 7)(a - 5)$
* Multiply $a \cdot a = a^2$
* Multiply $a \cdot -5 = -5a$
* Multiply $-7 \cdot a = -7a$
* Multiply $-7 \cdot -5 = 35$
* Combine like terms: $-5a - 7a = -12a$
* Result: $a^2 - 12a + 35$

4. $(3x - 4)(2x - 8)$
* Multiply $3x \cdot 2x = 6x^2$
* Multiply $3x \cdot -8 = -24x$
* Multiply $-4 \cdot 2x = -8x$
* Multiply $-4 \cdot -8 = 32$
* Combine like terms: $-24x - 8x = -32x$
* Result: $6x^2 - 32x + 32$

5. $(x - 2)(x + 1)$
* Multiply $x \cdot x = x^2$
* Multiply $x \cdot 1 = x$
* Multiply $-2 \cdot x = -2x$
* Multiply $-2 \cdot 1 = -2$
* Combine like terms: $x - 2x = -x$
* Result: $x^2 - x - 2$

6. $(x - 1)(x^2 + 6x - 1)$
* Distribute $x$: $x(x^2) + x(6x) + x(-1) = x^3 + 6x^2 - x$
* Distribute $-1$: $-1(x^2) - 1(6x) - 1(-1) = -x^2 - 6x + 1$
* Combine all terms: $x^3 + (6x^2 - x^2) + (-x - 6x) + 1$
* Simplify: $x^3 + 5x^2 - 7x + 1$
* Result: $x^3 + 5x^2 - 7x + 1$

7. $(x - 8)(x - 2)$
* Multiply $x \cdot x = x^2$
* Multiply $x \cdot -2 = -2x$
* Multiply $-8 \cdot x = -8x$
* Multiply $-8 \cdot -2 = 16$
* Combine like terms: $-2x - 8x = -10x$
* Result: $x^2 - 10x + 16$

8. $(x + 6)(x - 3)$
* Multiply $x \cdot x = x^2$
* Multiply $x \cdot -3 = -3x$
* Multiply $6 \cdot x = 6x$
* Multiply $6 \cdot -3 = -18$
* Combine like terms: $-3x + 6x = 3x$
* Result: $x^2 + 3x - 18$

9. $(4x + 1)(3x - 4)$
* Multiply $4x \cdot 3x = 12x^2$
* Multiply $4x \cdot -4 = -16x$
* Multiply $1 \cdot 3x = 3x$
* Multiply $1 \cdot -4 = -4$
* Combine like terms: $-16x + 3x = -13x$
* Result: $12x^2 - 13x - 4$

10. $(2x - 6)(x + 2)$
* Multiply $2x \cdot x = 2x^2$
* Multiply $2x \cdot 2 = 4x$
* Multiply $-6 \cdot x = -6x$
* Multiply $-6 \cdot 2 = -12$
* Combine like terms: $4x - 6x = -2x$
* Result: $2x^2 - 2x - 12$

Final Answer:
1. $a^2 + 11a + 28$
2. $a^2 - 10a + 21$
3. $a^2 - 12a + 35$
4. $6x^2 - 32x + 32$
5. $x^2 - x - 2$
6. $x^3 + 5x^2 - 7x + 1$
7. $x^2 - 10x + 16$
8. $x^2 + 3x - 18$
9. $12x^2 - 13x - 4$
10. $2x^2 - 2x - 12$