1. The problem is to solve the inequality $$5x^2 + 5x > 10$$. 2. First, rewrite the inequality by moving all terms to one side: $$5x^2 + 5x - 10 > 0$$ 3. Factor out the common factor 5: $$5(x^2 + x - 2) > 0$$ 4. Since 5 is positive, the inequality depends on the quadratic: $$x^2 + x - 2 > 0$$ 5. Factor the quadratic: $$x^2 + x - 2 = (x + 2)(x - 1)$$ 6. So the inequality becomes: $$(x + 2)(x - 1) > 0$$ 7. To solve $(x + 2)(x - 1) > 0$, find the critical points where the expression equals zero: $$x = -2, \quad x = 1$$ 8. Test intervals determined by these points: - For $x < -2$, choose $x = -3$: $( -3 + 2)( -3 - 1) = (-1)(-4) = 4 > 0$ (True) - For $-2 < x < 1$, choose $x = 0$: $(0 + 2)(0 - 1) = 2 \times (-1) = -2 < 0$ (False) - For $x > 1$, choose $x = 2$: $(2 + 2)(2 - 1) = 4 \times 1 = 4 > 0$ (True) 9. Therefore, the solution to the inequality is: $$x < -2 \quad \text{or} \quad x > 1$$ 10. Final answer: $$\boxed{x < -2 \text{ or } x > 1}$$