1. **State the problem:** Solve the equation $$a^3 + a^2 = 36$$ for the variable $a$. 2. **Rewrite the equation:** We want to find $a$ such that $$a^3 + a^2 - 36 = 0.$$ This is a cubic equation. 3. **Factor the equation if possible:** Factor out the common term $a^2$ from the first two terms: $$a^2(a + 1) - 36 = 0.$$ 4. **Isolate terms:** Rewrite as $$a^2(a + 1) = 36.$$ 5. **Try possible integer roots:** Since 36 is positive, try integer values for $a$ that satisfy the equation. 6. **Test $a=3$:** $$3^3 + 3^2 = 27 + 9 = 36,$$ which satisfies the equation. 7. **Test $a=-4$:** $$(-4)^3 + (-4)^2 = -64 + 16 = -48 eq 36,$$ so no. 8. **Check for other roots:** The cubic may have other roots, but since $a=3$ satisfies the equation and the problem likely expects real roots, $a=3$ is a valid solution. 9. **Summary:** The solution to $$a^3 + a^2 = 36$$ is $$\boxed{3}.$$