1. **State the problem:** Solve the quadratic equation by factoring: $$x^2 - 38 = -3x - 10$$ 2. **Rewrite the equation:** Move all terms to one side to set the equation equal to zero: $$x^2 - 38 + 3x + 10 = 0$$ 3. **Simplify:** Combine like terms: $$x^2 + 3x - 28 = 0$$ 4. **Factor the quadratic:** Find two numbers that multiply to $$-28$$ and add to $$3$$. These numbers are $$7$$ and $$-4$$. So, $$x^2 + 3x - 28 = (x + 7)(x - 4) = 0$$ 5. **Set each factor equal to zero:** $$x + 7 = 0 \quad \Rightarrow \quad x = -7$$ $$x - 4 = 0 \quad \Rightarrow \quad x = 4$$ 6. **Final answer:** $$\boxed{x = -7 \text{ or } x = 4}$$