Question: User: top-left: A fraction expression shows $\frac{6x^2 - x - 2}{2x^2 + 3x + 1}$, multiplied by $\frac{2x^2 + x - 1}{3x^2 + 10x - 8}$.\nbottom-left: A fraction expression shows $\frac{x^2 - 7x + 6}{x^2 + x - 2}$, divided by $\frac{x^2 - x - 12}{x^2 - 2x - 8}$.\ntop-right: b. A fraction expression shows $\frac{x^2 + 4x - 5}{x^2 + x - 2}$, divided by $\frac{x^2 + 10x + 25}{x^2 + 4x + 4}$.I want for all this to be explained, step by step
1. **Problem 1 (top-left): Simplify $$\frac{6x^2 - x - 2}{2x^2 + 3x + 1} \times \frac{2x^2 + x - 1}{3x^2 + 10x - 8}$$.
2. **Factor all polynomials:**
- $6x^2 - x - 2 = (3x + 2)(2x - 1)$
- $2x^2 + 3x + 1 = (2x + 1)(x + 1)$
- $2x^2 + x - 1 = (2x - 1)(x + 1)$
- $3x^2 + 10x - 8 = (3x - 2)(x + 4)$
3. **Rewrite the expression with factors:**
$$\frac{(3x + 2)(2x - 1)}{(2x + 1)(x + 1)} \times \frac{(2x - 1)(x + 1)}{(3x - 2)(x + 4)}$$
4. **Cancel common factors:**
- Cancel $(2x - 1)$ and $(x + 1)$ from numerator and denominator:
$$\frac{(3x + 2)\cancel{(2x - 1)}}{(2x + 1)\cancel{(x + 1)}} \times \frac{\cancel{(2x - 1)}\cancel{(x + 1)}}{(3x - 2)(x + 4)} = \frac{3x + 2}{2x + 1} \times \frac{1}{(3x - 2)(x + 4)}$$
5. **Multiply remaining factors:**
$$\frac{3x + 2}{2x + 1} \times \frac{1}{(3x - 2)(x + 4)} = \frac{3x + 2}{(2x + 1)(3x - 2)(x + 4)}$$
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6. **Problem 2 (bottom-left): Simplify $$\frac{x^2 - 7x + 6}{x^2 + x - 2} \div \frac{x^2 - x - 12}{x^2 - 2x - 8}$$.
7. **Factor all polynomials:**
- $x^2 - 7x + 6 = (x - 6)(x - 1)$
- $x^2 + x - 2 = (x + 2)(x - 1)$
- $x^2 - x - 12 = (x - 4)(x + 3)$
- $x^2 - 2x - 8 = (x - 4)(x + 2)$
8. **Rewrite division as multiplication by reciprocal:**
$$\frac{(x - 6)(x - 1)}{(x + 2)(x - 1)} \times \frac{(x - 4)(x + 2)}{(x - 4)(x + 3)}$$
9. **Cancel common factors:**
- Cancel $(x - 1)$, $(x + 2)$, and $(x - 4)$:
$$\frac{(x - 6)\cancel{(x - 1)}}{\cancel{(x + 2)}\cancel{(x - 1)}} \times \frac{\cancel{(x - 4)}\cancel{(x + 2)}}{\cancel{(x - 4)}(x + 3)} = \frac{x - 6}{1} \times \frac{1}{x + 3} = \frac{x - 6}{x + 3}$$
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10. **Problem 3 (top-right b): Simplify $$\frac{x^2 + 4x - 5}{x^2 + x - 2} \div \frac{x^2 + 10x + 25}{x^2 + 4x + 4}$$.
11. **Factor all polynomials:**
- $x^2 + 4x - 5 = (x + 5)(x - 1)$
- $x^2 + x - 2 = (x + 2)(x - 1)$
- $x^2 + 10x + 25 = (x + 5)^2$
- $x^2 + 4x + 4 = (x + 2)^2$
12. **Rewrite division as multiplication by reciprocal:**
$$\frac{(x + 5)(x - 1)}{(x + 2)(x - 1)} \times \frac{(x + 2)^2}{(x + 5)^2}$$
13. **Cancel common factors:**
- Cancel $(x - 1)$ and one $(x + 5)$ and one $(x + 2)$:
$$\frac{(x + 5)\cancel{(x - 1)}}{(x + 2)\cancel{(x - 1)}} \times \frac{(x + 2)\cancel{(x + 2)}}{(x + 5)\cancel{(x + 5)}} = \frac{x + 5}{x + 2} \times \frac{x + 2}{x + 5}$$
14. **Cancel remaining factors:**
$$\frac{\cancel{x + 5}}{\cancel{x + 2}} \times \frac{\cancel{x + 2}}{\cancel{x + 5}} = 1$$
**Final answers:**
- Problem 1: $$\frac{3x + 2}{(2x + 1)(3x - 2)(x + 4)}$$
- Problem 2: $$\frac{x - 6}{x + 3}$$
- Problem 3: $$1$$