1. **State the problem:** We want to find which function is equivalent to $f(x) = (x + 3)(x - 9)$ and matches the given handwritten expression $x^2 - 6x - 27$. 2. **Expand the original function:** Use the distributive property (FOIL) to expand: $$f(x) = (x + 3)(x - 9) = x^2 - 9x + 3x - 27 = x^2 - 6x - 27$$ 3. **Rewrite the quadratic in vertex form:** The vertex form is $$f(x) = a(x - h)^2 + k$$ where $(h,k)$ is the vertex. 4. **Complete the square:** Start with $$x^2 - 6x - 27$$ Take half of the coefficient of $x$, which is $-6$, half is $-3$, square it: $(-3)^2 = 9$. Add and subtract 9 inside the expression: $$x^2 - 6x + 9 - 9 - 27 = (x - 3)^2 - 36$$ 5. **Interpretation:** The vertex form is $$f(x) = (x - 3)^2 - 36$$ 6. **Match with options:** This matches option D. **Final answer:** $$\boxed{f(x) = (x - 3)^2 - 36}$$