1. **State the problem:** Simplify the expression involving two rational expressions: $$\frac{5}{(x-3)(x+1)} \quad \text{and} \quad \frac{x+1}{(x-2)(x+4)}$$ 2. **Understand the expressions:** Each is a fraction with polynomials in the denominator. 3. **Check for common factors:** The first denominator is $(x-3)(x+1)$ and the second denominator is $(x-2)(x+4)$. The numerators are $5$ and $x+1$ respectively. 4. **Simplify each expression if possible:** - The first expression $\frac{5}{(x-3)(x+1)}$ cannot be simplified further since 5 is a constant and the denominator factors are distinct. - The second expression $\frac{x+1}{(x-2)(x+4)}$ also cannot be simplified further since numerator and denominator share no common factors. 5. **If the problem is to add or subtract these rational expressions, find a common denominator:** - The common denominator is the product of all distinct factors: $(x-3)(x+1)(x-2)(x+4)$. 6. **Rewrite each fraction with the common denominator:** $$\frac{5}{(x-3)(x+1)} = \frac{5(x-2)(x+4)}{(x-3)(x+1)(x-2)(x+4)}$$ $$\frac{x+1}{(x-2)(x+4)} = \frac{(x+1)(x-3)(x+1)}{(x-3)(x+1)(x-2)(x+4)}$$ 7. **Simplify the numerator of the second fraction:** Note $(x+1)(x+1) = (x+1)^2$. 8. **Add the numerators:** $$5(x-2)(x+4) + (x+1)^2(x-3)$$ 9. **Expand each term:** - $5(x-2)(x+4) = 5(x^2 + 2x - 8) = 5x^2 + 10x - 40$ - $(x+1)^2(x-3) = (x^2 + 2x + 1)(x-3)$ 10. **Expand $(x^2 + 2x + 1)(x-3)$:** $$x^3 - 3x^2 + 2x^2 - 6x + x - 3 = x^3 - x^2 - 5x - 3$$ 11. **Add the two expanded numerators:** $$5x^2 + 10x - 40 + x^3 - x^2 - 5x - 3 = x^3 + (5x^2 - x^2) + (10x - 5x) + (-40 - 3) = x^3 + 4x^2 + 5x - 43$$ 12. **Final simplified expression:** $$\frac{x^3 + 4x^2 + 5x - 43}{(x-3)(x+1)(x-2)(x+4)}$$ **Answer:** $$\boxed{\frac{x^3 + 4x^2 + 5x - 43}{(x-3)(x+1)(x-2)(x+4)}}$$