1. **State the problem:** Simplify the expression $$\frac{x+8}{3x-1} + \frac{x+3}{x+1}$$. 2. **Find a common denominator:** The denominators are $3x-1$ and $x+1$. The common denominator is their product: $$(3x-1)(x+1)$$. 3. **Rewrite each fraction with the common denominator:** $$\frac{x+8}{3x-1} = \frac{(x+8)(x+1)}{(3x-1)(x+1)}$$ $$\frac{x+3}{x+1} = \frac{(x+3)(3x-1)}{(x+1)(3x-1)}$$ 4. **Add the numerators:** $$\frac{(x+8)(x+1) + (x+3)(3x-1)}{(3x-1)(x+1)}$$ 5. **Expand the numerators:** $$(x+8)(x+1) = x^2 + x + 8x + 8 = x^2 + 9x + 8$$ $$(x+3)(3x-1) = 3x^2 - x + 9x - 3 = 3x^2 + 8x - 3$$ 6. **Sum the expanded numerators:** $$x^2 + 9x + 8 + 3x^2 + 8x - 3 = 4x^2 + 17x + 5$$ 7. **Write the full expression:** $$\frac{4x^2 + 17x + 5}{(3x-1)(x+1)}$$ 8. **Factor the numerator if possible:** Try to factor $4x^2 + 17x + 5$. Find two numbers that multiply to $4 \times 5 = 20$ and add to $17$: these are $5$ and $4$. Rewrite: $$4x^2 + 5x + 12x + 5 = (4x^2 + 5x) + (12x + 5)$$ Factor each group: $$x(4x + 5) + 1(12x + 5)$$ Since the terms inside parentheses differ, try grouping differently or use quadratic formula. 9. **Use quadratic formula to check factorability:** $$x = \frac{-17 \pm \sqrt{17^2 - 4 \times 4 \times 5}}{2 \times 4} = \frac{-17 \pm \sqrt{289 - 80}}{8} = \frac{-17 \pm \sqrt{209}}{8}$$ Since $\sqrt{209}$ is irrational, numerator does not factor nicely. 10. **Final simplified form:** $$\boxed{\frac{4x^2 + 17x + 5}{(3x-1)(x+1)}}$$ This is the simplest form of the expression.