1. **State the problem:** Solve the equation $$\frac{L+2}{L-2} = \frac{-4}{L+2}$$ for $L$. 2. **Formula and rules:** To solve rational equations, we cross-multiply to eliminate denominators, then solve the resulting polynomial equation. 3. **Cross-multiply:** $$ (L+2)(L+2) = -4(L-2) $$ 4. **Expand both sides:** $$ (L+2)^2 = -4(L-2) $$ $$ L^2 + 4L + 4 = -4L + 8 $$ 5. **Bring all terms to one side:** $$ L^2 + 4L + 4 + 4L - 8 = 0 $$ $$ L^2 + 8L - 4 = 0 $$ 6. **Solve quadratic equation:** Use the quadratic formula $$ L = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $$ where $a=1$, $b=8$, $c=-4$. 7. **Calculate discriminant:** $$ \Delta = 8^2 - 4 \times 1 \times (-4) = 64 + 16 = 80 $$ 8. **Find roots:** $$ L = \frac{-8 \pm \sqrt{80}}{2} = \frac{-8 \pm 4\sqrt{5}}{2} $$ 9. **Simplify:** $$ L = -4 \pm 2\sqrt{5} $$ 10. **Check for restrictions:** Denominators cannot be zero, so $L \neq 2$ and $L \neq -2$. Both solutions satisfy this. **Final answer:** $$ L = -4 + 2\sqrt{5} \quad \text{or} \quad L = -4 - 2\sqrt{5} $$