1. **State the problem:** Solve the equation $$4 \{2 - [3(c + 1) - (c + 1)7]^2\} = -2c$$ for $c$. 2. **Simplify inside the brackets:** Calculate $3(c + 1) - (c + 1)7$. 3. Factor out $(c + 1)$: $$3(c + 1) - 7(c + 1) = (c + 1)(3 - 7) = (c + 1)(-4) = -4(c + 1)$$ 4. Substitute back: $$4 \{2 - [-4(c + 1)]^2\} = -2c$$ 5. Square the term inside the braces: $$[-4(c + 1)]^2 = 16(c + 1)^2$$ 6. So the equation becomes: $$4 \{2 - 16(c + 1)^2\} = -2c$$ 7. Distribute 4: $$8 - 64(c + 1)^2 = -2c$$ 8. Rearrange to isolate terms: $$8 + 2c = 64(c + 1)^2$$ 9. Expand $(c + 1)^2$: $$8 + 2c = 64(c^2 + 2c + 1)$$ 10. Distribute 64: $$8 + 2c = 64c^2 + 128c + 64$$ 11. Bring all terms to one side: $$0 = 64c^2 + 128c + 64 - 2c - 8$$ 12. Simplify: $$0 = 64c^2 + 126c + 56$$ 13. Divide entire equation by 2 to simplify: $$0 = \cancel{2}64c^2/\cancel{2} + \cancel{2}126c/\cancel{2} + \cancel{2}56/\cancel{2}$$ $$0 = 32c^2 + 63c + 28$$ 14. Solve quadratic equation $32c^2 + 63c + 28 = 0$ using quadratic formula: $$c = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a=32$, $b=63$, $c=28$. 15. Calculate discriminant: $$\Delta = 63^2 - 4 \times 32 \times 28 = 3969 - 3584 = 385$$ 16. Calculate roots: $$c = \frac{-63 \pm \sqrt{385}}{64}$$ **Final answer:** $$c = \frac{-63 + \sqrt{385}}{64} \quad \text{or} \quad c = \frac{-63 - \sqrt{385}}{64}$$