1. **State the problem:** Solve the equation $x^2 + 9x + 7 = -x^2 - 2$ for $x$. 2. **Bring all terms to one side:** Add $x^2$ and $2$ to both sides to combine like terms: $$x^2 + 9x + 7 + x^2 + 2 = 0$$ 3. **Simplify the equation:** $$2x^2 + 9x + 9 = 0$$ 4. **Use the quadratic formula:** For an equation $ax^2 + bx + c = 0$, the solutions are given by $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a=2$, $b=9$, and $c=9$. 5. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = 9^2 - 4 \times 2 \times 9 = 81 - 72 = 9$$ 6. **Find the roots:** $$x = \frac{-9 \pm \sqrt{9}}{2 \times 2} = \frac{-9 \pm 3}{4}$$ 7. **Calculate each root:** - For $+$ sign: $$x = \frac{-9 + 3}{4} = \frac{-6}{4} = -\frac{3}{2}$$ - For $-$ sign: $$x = \frac{-9 - 3}{4} = \frac{-12}{4} = -3$$ **Final answer:** The solutions are $x = -\frac{3}{2}$ and $x = -3$.