1. **State the problem:** Solve the equation $-4b (3b - 5b - 7) = -5 (2b + 9) + 8$ for $b$. 2. **Simplify inside parentheses:** $$3b - 5b - 7 = (3b - 5b) - 7 = -2b - 7$$ 3. **Rewrite the equation:** $$-4b(-2b - 7) = -5(2b + 9) + 8$$ 4. **Distribute terms:** $$-4b \times -2b = 8b^2, \quad -4b \times -7 = 28b$$ $$-5 \times 2b = -10b, \quad -5 \times 9 = -45$$ So the equation becomes: $$8b^2 + 28b = -10b - 45 + 8$$ 5. **Combine like terms on the right:** $$-10b - 45 + 8 = -10b - 37$$ 6. **Bring all terms to one side:** $$8b^2 + 28b + 10b + 37 = 0$$ 7. **Combine like terms:** $$8b^2 + 38b + 37 = 0$$ 8. **Use quadratic formula:** The quadratic formula is: $$b = \frac{-B \pm \sqrt{B^2 - 4AC}}{2A}$$ where $A=8$, $B=38$, and $C=37$. 9. **Calculate discriminant:** $$\Delta = B^2 - 4AC = 38^2 - 4 \times 8 \times 37 = 1444 - 1184 = 260$$ 10. **Calculate roots:** $$b = \frac{-38 \pm \sqrt{260}}{16}$$ 11. **Simplify square root:** $$\sqrt{260} = \sqrt{4 \times 65} = 2\sqrt{65}$$ 12. **Final solution:** $$b = \frac{-38 \pm 2\sqrt{65}}{16} = \frac{-19 \pm \sqrt{65}}{8}$$ **Answer:** $$b = \frac{-19 + \sqrt{65}}{8} \quad \text{or} \quad b = \frac{-19 - \sqrt{65}}{8}$$