1. **State the problem:** Solve for $a$ in the equation $$a^3 + a^2 = 36.$$\n\n2. **Rewrite the equation:** We want to find $a$ such that $$a^3 + a^2 - 36 = 0.$$\n\n3. **Factor the equation:** Factor out the common term $a^2$ from the first two terms:\n$$a^2(a + 1) - 36 = 0.$$\n\n4. **Isolate terms:** Rewrite as $$a^2(a + 1) = 36.$$\n\n5. **Try possible integer factors:** Since $36$ is positive, $a$ could be positive or negative. Test integer values to find roots.\n\n6. **Test $a=3$:** $$3^3 + 3^2 = 27 + 9 = 36,$$ which satisfies the equation.\n\n7. **Test $a=-4$:** $$(-4)^3 + (-4)^2 = -64 + 16 = -48 eq 36.$$\n\n8. **Conclusion:** The solution is $$a = 3.$$