Let f(x)= - X^2 - 4 x - 1 and g(x) = - 3 X^2 + 5 x + 4 . Find g(f(0)) .

f(x) = -x^2 - 4x -1

g(x) = -3x^2 +
5x + 4

find g(f(0)

First we need to
determine  g(f(x)

g(f(x) = g ( -x^2 - 4x
-1)

         = -3(x^2-4x-1)^2 + 5(-x^2 - 4x -1)  +
4

Let us factor (-x^2 - 4x -1)
:

        = ( -x^2 - 4x -1) [ -3(x^2 - 4x -1) + 5] +
4

         = ( -x^2 - 4x -1) ( -3x^2 +12x + 3 + 5) +
4

         = ( -x^2 - 4x -1) ( -3x^2 + 12x + 8)  +
4

Now we will substitute with x =
0:

==> g(f(0) = ( -0 -0 -1) ( -0+0 + 8)  +
4

                = -1* 8  +
4

                = -8 + 4 =
-4

==> g(f(0)) =
-4