1. **State the problem:** Factorize the expression completely: $$(x+3)^2 - 5(x^2 - 9)$$ 2. **Expand the terms:** $$(x+3)^2 = x^2 + 6x + 9$$ $$-5(x^2 - 9) = -5x^2 + 45$$ 3. **Combine like terms:** $$x^2 + 6x + 9 - 5x^2 + 45 = (x^2 - 5x^2) + 6x + (9 + 45) = -4x^2 + 6x + 54$$ 4. **Factor out the greatest common factor (GCF):** $$-4x^2 + 6x + 54 = -2(2x^2 - 3x - 27)$$ 5. **Factor the quadratic inside the parentheses:** We look for two numbers that multiply to $2 \times (-27) = -54$ and add to $-3$. These numbers are $6$ and $-9$. 6. **Rewrite and factor by grouping:** $$2x^2 - 3x - 27 = 2x^2 + 6x - 9x - 27$$ Group terms: $$(2x^2 + 6x) - (9x + 27) = 2x(x + 3) - 9(x + 3)$$ 7. **Factor out the common binomial:** $$(2x - 9)(x + 3)$$ 8. **Write the complete factorization:** $$-2(2x - 9)(x + 3)$$ **Final answer:** $$-2(2x - 9)(x + 3)$$