1. **State the problem:** Factor the quadratic expression $9a^2 + 9a + 36$. 2. **Recall the factoring formula:** For a quadratic $ax^2 + bx + c$, we look for factors of $a \times c$ that add up to $b$. 3. **Identify coefficients:** Here, $a=9$, $b=9$, and $c=36$. 4. **Calculate $a \times c$:** $$9 \times 36 = 324$$ 5. **Find factors of 324 that sum to 9:** Factors of 324 include 18 and 18, 27 and 12, 36 and 9, etc. None of these pairs add to 9. 6. **Check for common factors:** All terms have a common factor of 9. 7. **Factor out 9:** $$9a^2 + 9a + 36 = 9(a^2 + a + 4)$$ 8. **Try to factor $a^2 + a + 4$:** The discriminant is $$\Delta = b^2 - 4ac = 1^2 - 4 \times 1 \times 4 = 1 - 16 = -15$$ which is negative, so it cannot be factored over the real numbers. 9. **Final factored form:** $$9(a^2 + a + 4)$$ This is the simplest factorization over the real numbers.