1. **State the problem:** Simplify and analyze 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 factors as $$(a - 2b)^2$$. 3. **Factor the quadratic in $$x$$:** The terms $$-x^2 + 6x + 9$$ can be rewritten as $$-(x^2 - 6x - 9)$$. To complete the square for $$x^2 - 6x - 9$$: $$x^2 - 6x - 9 = (x^2 - 6x + 9) - 9 - 9 = (x - 3)^2 - 18$$ So, $$-x^2 + 6x + 9 = -[(x - 3)^2 - 18] = - (x - 3)^2 + 18$$ 4. **Rewrite the entire expression:** $$a^2 - 4ab + 4b^2 - x^2 + 6x + 9 = (a - 2b)^2 - (x - 3)^2 + 18$$ 5. **Interpretation:** The expression is the difference of two squares plus a constant. This form is useful for graphing or further algebraic manipulation. **Final simplified form:** $$\boxed{(a - 2b)^2 - (x - 3)^2 + 18}$$