Find the vertex of G(x) = x 2 + 4x.

If you mean  G(x)  =  x² + 4x:

(I'll use a 'y' instead of a 'G(x)'.)

Complete the square:     y + 4  =  x² + 4x + 4

Factor:                          y + 4  =  (x + 2)² 

Any equation in the form  y - k  =  (x - h)²  is a parabola with its vertex at (h, k);

therefore, for this equation, the vertex is at  (-2, -4).

do you mean G(x) = x² + 4x ?

If not. There is no vertex in a linear function.