1. **Problem statement:** Solve the quadratic equation $$x^2 + 8x - 100 = -x^2 - 40 + 10x$$. 2. **Step 1: Bring all terms to one side to set the equation to zero.** $$x^2 + 8x - 100 + x^2 + 40 - 10x = 0$$ 3. **Step 2: Combine like terms.** $$x^2 + x^2 + 8x - 10x - 100 + 40 = 0$$ $$2x^2 - 2x - 60 = 0$$ 4. **Step 3: Simplify the equation by dividing all terms by 2.** $$\cancel{2}x^2 - \cancel{2}x - \cancel{60} = 0 \Rightarrow x^2 - x - 30 = 0$$ 5. **Step 4: Use the quadratic formula to solve for $x$.** The quadratic formula is: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a=1$, $b=-1$, and $c=-30$. 6. **Step 5: Calculate the discriminant.** $$\Delta = b^2 - 4ac = (-1)^2 - 4(1)(-30) = 1 + 120 = 121$$ 7. **Step 6: Calculate the roots.** $$x = \frac{-(-1) \pm \sqrt{121}}{2(1)} = \frac{1 \pm 11}{2}$$ 8. **Step 7: Find the two solutions.** - For the plus sign: $$x = \frac{1 + 11}{2} = \frac{12}{2} = 6$$ - For the minus sign: $$x = \frac{1 - 11}{2} = \frac{-10}{2} = -5$$ **Final answer:** $$x = 6 \quad \text{or} \quad x = -5$$