Step 1 :Find the composition of the functions: \(f(g(x))\)
Step 2 :\(f(g(x)) = f(x^2 + 2x + 2)\)
Step 3 :\(f(g(x)) = (x^2 + 2x + 2)^2 - 2(x^2 + 2x + 2) - 5\)
Step 4 :Substitute \(x = -4\) into the composed function: \(f(g(-4))\)
Step 5 :\(f(g(-4)) = ((-4)^2 + 2(-4) + 2)^2 - 2((-4)^2 + 2(-4) + 2) - 5\)
Step 6 :\(f(g(-4)) = (16 - 8 + 2)^2 - 2(16 - 8 + 2) - 5\)
Step 7 :\(f(g(-4)) = (10)^2 - 2(10) - 5\)
Step 8 :\(f(g(-4)) = 100 - 20 - 5\)
Step 9 :\(f(g(-4)) = 75\)
Step 10 :\(\boxed{75}\)