Given the function $f(x)=2 x^{2}-5 x+8$. Calculate the following values using synthetic division and the Remainder Theorem:
\[
\begin{array}{l}
f(-2)= \\
f(-1)= \\
f(0)= \\
f(1)= \\
f(2)=
\end{array}
\]

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Step 1 ：Given the function \(f(x)=2 x^{2}-5 x+8\). We need to calculate the following values using synthetic division and the Remainder Theorem: \(f(-2)\), \(f(-1)\), \(f(0)\), \(f(1)\), \(f(2)\).

Step 2 ：The Remainder Theorem states that if a polynomial f(x) is divided by (x-a), then the remainder is f(a). This means that to find the value of the function at a certain point, we can simply substitute that point into the function.

Step 3 ：Substitute -2, -1, 0, 1, and 2 into the function to find the corresponding values.

Step 4 ：The values of the function \(f(x)=2 x^{2}-5 x+8\) at \(x=-2, -1, 0, 1, 2\) are 26, 15, 8, 5, 6 respectively.

Step 5 ：So, we have: \[\begin{array}{l} f(-2)= \boxed{26} \\ f(-1)= \boxed{15} \\ f(0)= \boxed{8} \\ f(1)= \boxed{5} \\ f(2)= \boxed{6} \end{array}\]