Step 1 :Let's first find the value of \(g(-3)\). We substitute \(x = -3\) into the function \(g(x) = x^{2} + 5\).
Step 2 :After calculating, we find that \(g(-3) = 14\).
Step 3 :Next, we substitute this value into the function \(f(x) = 4x - 1\).
Step 4 :After calculating, we find that \(f(14) = 55\).
Step 5 :So, the value of \((f \circ g)(-3)\) is \(\boxed{55}\).