Complete the square of the given quadratic expression. Then, graph the function using the technique of shifting
\[
f(x)=-2 x^{2}-16 x-31
\]

Complete the square by entering the correct numbers into the expression below
\[
f(x)=\square(x+\square)^{2}+
\]

Graph $f(x)$. Use the graphing tool provided to graph the function
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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 ：\(\boxed{f(x)=-2(x+4)^{2}+1}\)