Find the inverse Laplace transform $f(t)=\mathcal{L}^{-1}\{F(s)\}$ of the function
\[
F(s)=\frac{s-5}{s^{2}-6 s+25}
\]
\[
f(t)=\mathcal{L}^{-1}\left\{\frac{s-5}{s^{2}-6 s+25}\right\}=
\]

Step 1 ：We are given the function \(F(s)=\frac{s-5}{s^{2}-6 s+25}\) and we are asked to find its inverse Laplace transform.

Step 2 ：The denominator of the function can be factored as \((s-3)^2 + 16\). This suggests that the function can be written in the form of \(A(s-3) + B\), where A and B are constants.

Step 3 ：We can find the constants A and B by comparing coefficients. After solving, we find that A = 1 and B = -8.

Step 4 ：Now, we can write the function in the form of \(A(s-3) + B\) and use the standard formula for inverse Laplace transform.

Step 5 ：The inverse Laplace transform of the form \(\frac{s-a}{(s-a)^2 + b^2}\) and \(\frac{b}{(s-a)^2 + b^2}\) are \(e^{at}cos(bt)\) and \(e^{at}sin(bt)\) respectively.

Step 6 ：Applying these formulas, we find that the inverse Laplace transform of the function \(F(s)=\frac{s-5}{s^{2}-6 s+25}\) is \(f(t) = -2e^{3t}sin(4t) + e^{3t}cos(4t)\) for \(t > 0\).

Step 7 ：\(\boxed{f(t) = -2e^{3t}sin(4t) + e^{3t}cos(4t)}\) is the final answer.