Let's go through the simplification of each expression, one by one.

Problem 3

$\frac{2x^2}{3x^2 - 4x}$

1. Factor the denominator: $3x^2 - 4x = x(3x - 4)$.
2. Simplify by canceling $x$ (if $x \neq 0$): $\frac{2x^2}{x(3x - 4)} = \frac{2x}{3x - 4}$

Problem 4

$\frac{7x^3 - x^2}{2x^3}$

1. Factor the numerator: $7x^3 - x^2 = x^2(7x - 1)$.
2. Simplify by canceling $x^2$ (if $x \neq 0$): $\frac{x^2(7x - 1)}{2x^3} = \frac{7x - 1}{2x}$

Problem 5

$\frac{x^2 - 3x - 18}{x^2 - 7x + 6}$

1. Factor both the numerator and denominator:
   • Numerator: $x^2 - 3x - 18 = (x - 6)(x + 3)$
   • Denominator: $x^2 - 7x + 6 = (x - 6)(x - 1)$
2. Simplify by canceling $(x - 6)$: $\frac{(x - 6)(x + 3)}{(x - 6)(x - 1)} = \frac{x + 3}{x - 1}$

Problem 6

$\frac{x^2 + 13x + 36}{x^2 - 7x + 10}$

1. Factor both the numerator and denominator:
   • Numerator: $x^2 + 13x + 36 = (x + 9)(x + 4)$
   • Denominator: $x^2 - 7x + 10 = (x - 5)(x - 2)$
2. Since there are no common factors, the expression remains: $\frac{(x + 9)(x + 4)}{(x - 5)(x - 2)}$

Problem 7

$\frac{x^2 + 11x + 18}{x^3 + 8}$

1. Factor the numerator and denominator:
   • Numerator: $x^2 + 11x + 18 = (x + 9)(x + 2)$
   • Denominator: $x^3 + 8 = (x + 2)(x^2 - 2x + 4)$ (sum of cubes)
2. Simplify by canceling $(x + 2)$: $\frac{(x + 9)(x + 2)}{(x + 2)(x^2 - 2x + 4)} = \frac{x + 9}{x^2 - 2x + 4}$

Problem 8

$\frac{x^2 - 7x + 12}{x^3 - 27}$

1. Factor both the numerator and denominator:
   • Numerator: $x^2 - 7x + 12 = (x - 3)(x - 4)$
   • Denominator: $x^3 - 27 = (x - 3)(x^2 + 3x + 9)$ (difference of cubes)
2. Simplify by canceling $(x - 3)$: $\frac{(x - 3)(x - 4)}{(x - 3)(x^2 + 3x + 9)} = \frac{x - 4}{x^2 + 3x + 9}$

Problem 9

$\frac{32x^4 - 50}{4x^3 - 12x^2 - 5x + 15}$

1. Factor out common terms where possible:
   • Numerator: $32x^4 - 50 = 2(16x^4 - 25) = 2(4x^2 - 5)(4x^2 + 5)$
   • For the denominator, factor by grouping:
     • $4x^3 - 12x^2 - 5x + 15 = 4x^2(x - 3) - 5(x - 3) = (4x^2 - 5)(x - 3)$
2. The simplified form is: $\frac{2(4x^2 - 5)(4x^2 + 5)}{(4x^2 - 5)(x - 3)} = \frac{2(4x^2 + 5)}{x - 3}$

Problem 10

$\frac{3x^3 - 3x^2 + 7x - 7}{27x^4 - 147}$

1. Factor out common terms:
   • Numerator: $3x^3 - 3x^2 + 7x - 7 = 3x^2(x - 1) + 7(x - 1) = (3x^2 + 7)(x - 1)$
   • Denominator: (27x^4 - 147 = 27(x^4 - 5) = 27(x^2 - \sqrt{5})(x^