1. **State the problem:** Solve the equation $4x^2 - 5(x - 2) = 10$ for $x$. 2. **Expand and simplify:** Distribute the $-5$ across $(x - 2)$: $$4x^2 - 5x + 10 = 10$$ 3. **Bring all terms to one side:** Subtract $10$ from both sides: $$4x^2 - 5x + 10 - 10 = 10 - 10$$ $$4x^2 - 5x + \cancel{10} - \cancel{10} = \cancel{10} - \cancel{10}$$ $$4x^2 - 5x = 0$$ 4. **Factor the equation:** Factor out the common term $x$: $$x(4x - 5) = 0$$ 5. **Apply zero product property:** Set each factor equal to zero: $$x = 0$$ $$4x - 5 = 0$$ 6. **Solve for $x$ in the second equation:** $$4x = 5$$ $$\cancel{4}x = \cancel{5}$$ $$x = \frac{5}{4}$$ **Final answer:** The solutions are $x = 0$ and $x = \frac{5}{4}$.