1. **State the problem:** Solve for $a$ in the equation $$a^3 + a^2 = 36.$$\n\n2. **Rewrite the equation:** Move all terms to one side to set the equation to zero:\n$$a^3 + a^2 - 36 = 0.$$\n\n3. **Factor the equation if possible:** Factor out the common term $a^2$ from the first two terms:\n$$a^2(a + 1) - 36 = 0.$$\n\n4. **Isolate the constant:** Add 36 to both sides:\n$$a^2(a + 1) = 36.$$\n\n5. **Try possible integer roots:** Since 36 is a perfect square and cube, test integer values for $a$ such as 2 or 3.\n- For $a=2$: $$2^3 + 2^2 = 8 + 4 = 12 \neq 36.$$\n- For $a=3$: $$3^3 + 3^2 = 27 + 9 = 36,$$ which satisfies the equation.\n\n6. **Check for other roots:** The equation is cubic, so there may be other roots. Use the Rational Root Theorem or numerical methods to find others.\n\n7. **Summary:** The real root found is $$a = 3.$$\n\n**Final answer:** $$\boxed{3}.$$