1. **State the problem:** Solve the equation $$\frac{8}{x + 2} - \frac{5}{3} = \frac{x - 1}{3}$$ for $x$. 2. **Identify the formula and rules:** To solve rational equations, first find a common denominator to eliminate fractions by multiplying both sides. 3. **Find the common denominator:** The denominators are $x+2$ and $3$. The common denominator is $3(x+2)$. 4. **Multiply both sides by the common denominator:** $$3(x+2) \times \left(\frac{8}{x + 2} - \frac{5}{3}\right) = 3(x+2) \times \frac{x - 1}{3}$$ 5. **Simplify each term:** $$3(x+2) \times \frac{8}{x + 2} = 3 \times 8 = 24$$ $$3(x+2) \times \frac{5}{3} = (x+2) \times 5 = 5(x+2)$$ $$3(x+2) \times \frac{x - 1}{3} = (x+2)(x-1)$$ 6. **Rewrite the equation:** $$24 - 5(x+2) = (x+2)(x-1)$$ 7. **Expand terms:** $$24 - 5x - 10 = x^2 + 2x - x - 2$$ Simplify: $$24 - 5x - 10 = x^2 + x - 2$$ $$14 - 5x = x^2 + x - 2$$ 8. **Bring all terms to one side:** $$0 = x^2 + x - 2 - 14 + 5x$$ $$0 = x^2 + 6x - 16$$ 9. **Solve the quadratic equation:** Use the quadratic formula: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a=1$, $b=6$, $c=-16$. Calculate discriminant: $$\Delta = 6^2 - 4 \times 1 \times (-16) = 36 + 64 = 100$$ Calculate roots: $$x = \frac{-6 \pm \sqrt{100}}{2} = \frac{-6 \pm 10}{2}$$ 10. **Find the two solutions:** $$x_1 = \frac{-6 + 10}{2} = \frac{4}{2} = 2$$ $$x_2 = \frac{-6 - 10}{2} = \frac{-16}{2} = -8$$ 11. **Check for restrictions:** The denominator $x+2$ cannot be zero, so $x \neq -2$. Both $2$ and $-8$ are valid. **Final answer:** $$x = 2, -8$$