First, substitute \(f(x) = 2x + 3\) into \(g(x) = x^2 - 2x\):

\(gf(x) = (2x + 3)^2 - 2(2x + 3)\).

Expand \((2x + 3)^2\):

\((2x + 3)^2 = 4x^2 + 12x + 9\).

Expand \(-2(2x + 3)\):

\(-2(2x + 3) = -4x - 6\).

Combine the expressions:

\(gf(x) = 4x^2 + 12x + 9 - 4x - 6\).

Simplify:

\(gf(x) = 4x^2 + 8x + 3\).

Complete the square for \(4x^2 + 8x + 3\):

Factor out 4 from the quadratic terms:

\(4(x^2 + 2x) + 3\).

Complete the square inside the parentheses:

\(x^2 + 2x = (x + 1)^2 - 1\).

Substitute back:

\(4((x + 1)^2 - 1) + 3\).

Expand and simplify:

\(4(x + 1)^2 - 4 + 3 = 4(x + 1)^2 - 1\).

Thus, \(gf(x) = 4(x + 1)^2 - 1\).