EXPLANATION
Given the function f(x) = x^2 + 12x + 27
The expression in vertex form is as follows:
[tex]f(x)=a(x-h)^2+k[/tex]where a is the leading coefficient --> a=1
We need to compute h and k wich are the coordinates of the vertex (h,k)
The coordinates of the vertex are given by the following expression:
[tex]x_{\text{vertex}}=\frac{-b}{2a}[/tex]where b=12:
[tex]x_v=\frac{-12}{2}=-6[/tex]As x_v=-6 the expression of the function in standard form would be:
[tex]f(x)=(x-(-6))^2+k[/tex]Removing the parentheses:
[tex]f(x)=(x+6)^2+k[/tex]The only option that display this vertex form is the second one, hence that is the solution:
[tex]f(x)=(x-6)^2-9[/tex]