1. **State the problem:** Solve the equation $25(x + 2)^2 = (x - 7)^2 - 81$ for $x$. 2. **Rewrite the equation:** $$25(x + 2)^2 = (x - 7)^2 - 81$$ 3. **Expand both squares:** $$25(x^2 + 4x + 4) = (x^2 - 14x + 49) - 81$$ 4. **Distribute and simplify:** $$25x^2 + 100x + 100 = x^2 - 14x + (49 - 81)$$ $$25x^2 + 100x + 100 = x^2 - 14x - 32$$ 5. **Bring all terms to one side:** $$25x^2 + 100x + 100 - x^2 + 14x + 32 = 0$$ $$24x^2 + 114x + 132 = 0$$ 6. **Simplify by dividing all terms by 6:** $$\cancel{6}4x^2 + \cancel{6}19x + \cancel{6}22 = 0$$ $$4x^2 + 19x + 22 = 0$$ 7. **Use the quadratic formula:** $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a=4$, $b=19$, $c=22$. 8. **Calculate the discriminant:** $$\Delta = 19^2 - 4 \times 4 \times 22 = 361 - 352 = 9$$ 9. **Find the roots:** $$x = \frac{-19 \pm \sqrt{9}}{2 \times 4} = \frac{-19 \pm 3}{8}$$ 10. **Calculate each solution:** - $$x_1 = \frac{-19 + 3}{8} = \frac{-16}{8} = -2$$ - $$x_2 = \frac{-19 - 3}{8} = \frac{-22}{8} = -\frac{11}{4}$$ **Final answer:** $$x = -2 \quad \text{or} \quad x = -\frac{11}{4}$$