Step 1 :The given quadratic expression is \(f(x)=-2x^2-16x-31\).
Step 2 :The coefficient of \(x^2\) is \(-2\).
Step 3 :To find the value that completes the square, we can use the formula \(h=-\frac{b}{2a}\), where \(b\) is the coefficient of \(x\). In this case, \(b=-16\) and \(a=-2\).
Step 4 :After finding \(h\), we can substitute it back into the equation to find \(k\).
Step 5 :Finally, we can graph the function using the completed square form.
Step 6 :The completed square form of the given quadratic expression is \(f(x)=-2(x+4)^{2}+1\).
Step 7 :The graph of the function is a downward-opening parabola with vertex at \((-4,1)\).
Step 8 :\(\boxed{f(x)=-2(x+4)^{2}+1}\)