1. **State the problem:** Solve the equation $$a^3 + a^2 = 42a$$ for all values of $a$. 2. **Rewrite the equation:** Move all terms to one side to set the equation equal to zero: $$a^3 + a^2 - 42a = 0$$ 3. **Factor the equation:** Factor out the common factor $a$: $$a(a^2 + a - 42) = 0$$ 4. **Apply the zero product property:** For the product to be zero, either $$a = 0$$ or $$a^2 + a - 42 = 0$$ 5. **Solve the quadratic equation:** Use the quadratic formula for $$a^2 + a - 42 = 0$$ where $a=1$, $b=1$, and $c=-42$: $$a = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{-1 \pm \sqrt{1^2 - 4 \times 1 \times (-42)}}{2 \times 1} = \frac{-1 \pm \sqrt{1 + 168}}{2} = \frac{-1 \pm \sqrt{169}}{2}$$ 6. **Simplify the square root:** $$\sqrt{169} = 13$$ 7. **Find the two roots:** $$a = \frac{-1 + 13}{2} = \frac{12}{2} = 6$$ $$a = \frac{-1 - 13}{2} = \frac{-14}{2} = -7$$ 8. **List all solutions:** $$a = 0, 6, -7$$ **Final answer:** The solutions to the equation are $0, 6, -7$.