1. **State the problem:** Solve for $w$ in the equation $$-\frac{5}{w+2} = -8 - \frac{7}{w-1}.$$\n\n2. **Rewrite the equation:** Move all terms to one side to isolate fractions:\n$$-\frac{5}{w+2} + \frac{7}{w-1} = -8.$$\n\n3. **Find a common denominator:** The denominators are $w+2$ and $w-1$. The common denominator is $(w+2)(w-1)$.\n\n4. **Rewrite each fraction with the common denominator:**\n$$-\frac{5(w-1)}{(w+2)(w-1)} + \frac{7(w+2)}{(w+2)(w-1)} = -8.$$\n\n5. **Combine the fractions:**\n$$\frac{-5(w-1) + 7(w+2)}{(w+2)(w-1)} = -8.$$\n\n6. **Simplify the numerator:**\n$$-5(w-1) + 7(w+2) = -5w + 5 + 7w + 14 = 2w + 19.$$\n\n7. **Rewrite the equation:**\n$$\frac{2w + 19}{(w+2)(w-1)} = -8.$$\n\n8. **Cross-multiply:**\n$$2w + 19 = -8(w+2)(w-1).$$\n\n9. **Expand the right side:**\n$$(w+2)(w-1) = w^2 - w + 2w - 2 = w^2 + w - 2.$$\nSo,\n$$2w + 19 = -8(w^2 + w - 2).$$\n\n10. **Distribute -8:**\n$$2w + 19 = -8w^2 - 8w + 16.$$\n\n11. **Bring all terms to one side:**\n$$8w^2 + 2w + 8w + 19 - 16 = 0,$$\nwhich simplifies to\n$$8w^2 + 10w + 3 = 0.$$\n\n12. **Solve the quadratic equation:**\nUse the quadratic formula $$w = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a=8$, $b=10$, $c=3$.\n\nCalculate the discriminant:\n$$\Delta = 10^2 - 4 \times 8 \times 3 = 100 - 96 = 4.$$\n\n13. **Find the roots:**\n$$w = \frac{-10 \pm \sqrt{4}}{2 \times 8} = \frac{-10 \pm 2}{16}.$$\n\n14. **Calculate each solution:**\n- $$w_1 = \frac{-10 + 2}{16} = \frac{-8}{16} = -\frac{1}{2}.$$\n- $$w_2 = \frac{-10 - 2}{16} = \frac{-12}{16} = -\frac{3}{4}.$$\n\n15. **Check for restrictions:**\nThe original denominators cannot be zero, so $w \neq -2$ and $w \neq 1$. Both solutions are valid.\n\n**Final answer:** $$w = -\frac{1}{2}, -\frac{3}{4}.$$