1. **State the problem:** Solve for $x$ in the equation $$\frac{5}{x+1} + \frac{4}{3} = \frac{x+1}{x-1}.$$\n\n2. **Identify the formula and rules:** To solve rational equations, find a common denominator or multiply both sides by the least common denominator (LCD) to clear fractions. Remember to check for values that make denominators zero, as these are excluded.\n\n3. **Find the LCD:** The denominators are $x+1$, $3$, and $x-1$. The LCD is $$3(x+1)(x-1).$$\n\n4. **Multiply both sides by the LCD to clear fractions:**\n$$3(x+1)(x-1) \times \left( \frac{5}{x+1} + \frac{4}{3} \right) = 3(x+1)(x-1) \times \frac{x+1}{x-1}.$$\n\n5. **Distribute multiplication:**\n$$3(x-1) \times 5 + (x+1)(x-1) \times 4 = 3(x+1)^2.$$\n\n6. **Simplify each term:**\n$$15(x-1) + 4(x+1)(x-1) = 3(x+1)^2.$$\n\n7. **Expand products:**\n- $15(x-1) = 15x - 15$\n- $(x+1)(x-1) = x^2 - 1$\n- $3(x+1)^2 = 3(x^2 + 2x + 1) = 3x^2 + 6x + 3$\n\nSo the equation becomes:\n$$15x - 15 + 4(x^2 - 1) = 3x^2 + 6x + 3.$$\n\n8. **Expand and combine like terms:**\n$$15x - 15 + 4x^2 - 4 = 3x^2 + 6x + 3.$$\n$$4x^2 + 15x - 19 = 3x^2 + 6x + 3.$$\n\n9. **Bring all terms to one side:**\n$$4x^2 + 15x - 19 - 3x^2 - 6x - 3 = 0,$$\nwhich simplifies to\n$$x^2 + 9x - 22 = 0.$$\n\n10. **Solve the quadratic equation:**\nUse the quadratic formula $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ with $a=1$, $b=9$, $c=-22$.\n\nCalculate the discriminant:\n$$\Delta = 9^2 - 4 \times 1 \times (-22) = 81 + 88 = 169.$$\n\nSo,\n$$x = \frac{-9 \pm \sqrt{169}}{2} = \frac{-9 \pm 13}{2}.$$\n\n11. **Find the two solutions:**\n- $$x = \frac{-9 + 13}{2} = \frac{4}{2} = 2,$$\n- $$x = \frac{-9 - 13}{2} = \frac{-22}{2} = -11.$$\n\n12. **Check for excluded values:**\nDenominators $x+1$ and $x-1$ cannot be zero, so $x \neq -1$ and $x \neq 1$. Both $2$ and $-11$ are valid.\n\n**Final answer:** $$x = 2 \text{ or } x = -11.$$