1. State the problem: Solve for $a$ in the equation $a^3+a^2=36$. 2. Factor the left side using the common factor $a^2$: $$a^3+a^2=a^2(a+1)$$ 3. Rewrite the equation in factored form: $$a^2(a+1)=36$$ 4. Test values of $a$ that make the product $a^2(a+1)$ equal $36$ (note that $a^2\ge 0$): 5. Try $a=3$: $$a^2(a+1)=3^2(3+1)=9\cdot 4=36$$ 6. Try $a=-3$: $$a^2(a+1)=(-3)^2(-3+1)=9\cdot(-2)=-18\ne 36$$ 7. Try $a=-2$: $$a^2(a+1)=(-2)^2(-2+1)=4\cdot(-1)=-4\ne 36$$ 8. Conclude the only solution found is $a=3$. 9. Final answer: $a=3$.