1. **State the problem:** Simplify the given fractions: $$\frac{x + 1}{4x + 4}, \quad \frac{4x - 32}{x^2 - 5x - 24}, \quad \frac{x - 3}{5x - 15}, \quad \frac{x^2 + 3x - 18}{x^2 + 12x + 36}, \quad \frac{2x + 12}{x^2 + 14x + 48}, \quad \frac{x^2 + 4x - 32}{x^2 - 6x + 8}$$ 2. **Recall the formula and rules:** To simplify a fraction, factor numerator and denominator completely and cancel common factors. 3. **Simplify each fraction step-by-step:** **(a) Simplify** $\frac{x + 1}{4x + 4}$: Factor denominator: $$4x + 4 = 4(x + 1)$$ Fraction becomes: $$\frac{x + 1}{4(x + 1)}$$ Cancel common factor $x + 1$: $$\frac{\cancel{x + 1}}{4\cancel{(x + 1)}} = \frac{1}{4}$$ **(b) Simplify** $\frac{4x - 32}{x^2 - 5x - 24}$: Factor numerator: $$4x - 32 = 4(x - 8)$$ Factor denominator: $$x^2 - 5x - 24 = (x - 8)(x + 3)$$ Fraction becomes: $$\frac{4(x - 8)}{(x - 8)(x + 3)}$$ Cancel common factor $x - 8$: $$\frac{4\cancel{(x - 8)}}{\cancel{(x - 8)}(x + 3)} = \frac{4}{x + 3}$$ **(c) Simplify** $\frac{x - 3}{5x - 15}$: Factor denominator: $$5x - 15 = 5(x - 3)$$ Fraction becomes: $$\frac{x - 3}{5(x - 3)}$$ Cancel common factor $x - 3$: $$\frac{\cancel{x - 3}}{5\cancel{(x - 3)}} = \frac{1}{5}$$ **(d) Simplify** $\frac{x^2 + 3x - 18}{x^2 + 12x + 36}$: Factor numerator: $$x^2 + 3x - 18 = (x + 6)(x - 3)$$ Factor denominator: $$x^2 + 12x + 36 = (x + 6)^2$$ Fraction becomes: $$\frac{(x + 6)(x - 3)}{(x + 6)(x + 6)}$$ Cancel common factor $x + 6$: $$\frac{\cancel{(x + 6)}(x - 3)}{\cancel{(x + 6)}(x + 6)} = \frac{x - 3}{x + 6}$$ **(e) Simplify** $\frac{2x + 12}{x^2 + 14x + 48}$: Factor numerator: $$2x + 12 = 2(x + 6)$$ Factor denominator: $$x^2 + 14x + 48 = (x + 6)(x + 8)$$ Fraction becomes: $$\frac{2(x + 6)}{(x + 6)(x + 8)}$$ Cancel common factor $x + 6$: $$\frac{2\cancel{(x + 6)}}{\cancel{(x + 6)}(x + 8)} = \frac{2}{x + 8}$$ **(f) Simplify** $\frac{x^2 + 4x - 32}{x^2 - 6x + 8}$: Factor numerator: $$x^2 + 4x - 32 = (x + 8)(x - 4)$$ Factor denominator: $$x^2 - 6x + 8 = (x - 4)(x - 2)$$ Fraction becomes: $$\frac{(x + 8)(x - 4)}{(x - 4)(x - 2)}$$ Cancel common factor $x - 4$: $$\frac{(x + 8)\cancel{(x - 4)}}{\cancel{(x - 4)}(x - 2)} = \frac{x + 8}{x - 2}$$ 4. **Final simplified fractions:** $$\frac{1}{4}, \quad \frac{4}{x + 3}, \quad \frac{1}{5}, \quad \frac{x - 3}{x + 6}, \quad \frac{2}{x + 8}, \quad \frac{x + 8}{x - 2}$$