1. **State the problem:** Simplify the expression $$\frac{x^2 + 8x + 16}{x^2 - x - 20}$$ and state any restrictions on the variable. 2. **Factor numerator and denominator:** - Numerator: $$x^2 + 8x + 16 = (x + 4)(x + 4) = (x + 4)^2$$ - Denominator: $$x^2 - x - 20$$. Find two numbers that multiply to $$-20$$ and add to $$-1$$: these are $$-5$$ and $$4$$. So, $$x^2 - x - 20 = (x - 5)(x + 4)$$. 3. **Rewrite the expression:** $$\frac{(x + 4)^2}{(x - 5)(x + 4)}$$ 4. **Cancel common factors:** $$\frac{\cancel{(x + 4)}(x + 4)}{(x - 5)\cancel{(x + 4)}} = \frac{x + 4}{x - 5}$$ 5. **State restrictions:** - The denominator cannot be zero, so $$x - 5 \neq 0 \Rightarrow x \neq 5$$. - Also, the canceled factor $$x + 4$$ cannot be zero in the original expression, so $$x + 4 \neq 0 \Rightarrow x \neq -4$$. 6. **Final answer:** $$\frac{x + 4}{x - 5}$$ with restrictions $$x \neq 5$$ and $$x \neq -4$$.