1. **State the problem:** Factorize the expression $$a^2 - 4ab + 4b^2 - x^2 + 6x + 9$$. 2. **Recognize patterns:** The first three terms $$a^2 - 4ab + 4b^2$$ form a perfect square trinomial which can be written as $$(a - 2b)^2$$. 3. **Rewrite the expression:** $$ (a - 2b)^2 - x^2 + 6x + 9 $$ 4. **Focus on the remaining terms:** $$- x^2 + 6x + 9$$ can be rewritten by factoring out the negative sign: $$ -(x^2 - 6x - 9) $$ 5. **Try to factor or complete the square for $$x^2 - 6x - 9$$:** Complete the square: $$ x^2 - 6x - 9 = (x^2 - 6x + 9) - 18 = (x - 3)^2 - 18 $$ 6. **Substitute back:** $$ (a - 2b)^2 - [(x - 3)^2 - 18] = (a - 2b)^2 - (x - 3)^2 + 18 $$ 7. **Group terms:** $$ [(a - 2b)^2 - (x - 3)^2] + 18 $$ 8. **Use difference of squares formula:** $$ (p^2 - q^2) = (p - q)(p + q) $$ where $$p = (a - 2b)$$ and $$q = (x - 3)$$. So, $$ (a - 2b - (x - 3))(a - 2b + (x - 3)) + 18 $$ 9. **Simplify inside the parentheses:** $$ (a - 2b - x + 3)(a - 2b + x - 3) + 18 $$ 10. **Final factorized form:** The expression cannot be factored further nicely due to the +18 term. **Answer:** $$ (a - 2b - x + 3)(a - 2b + x - 3) + 18 $$