Express $f(x)$ in the form $f(x)=(x-k) q(x)+r$ for the given value of $k$.
\[
f(x)=5 x^{3}+x^{2}+x-7, k=-1
\]

Step 1 ：We are given the polynomial function \(f(x) = 5x^3 + x^2 + x - 7\) and the value \(k = -1\).

Step 2 ：We are asked to express the polynomial \(f(x)\) in the form \(f(x) = (x - k)q(x) + r\), which is equivalent to dividing the polynomial \(f(x)\) by the binomial \((x - k)\) and expressing the result as a quotient \(q(x)\) and a remainder \(r\).

Step 3 ：In this case, we are dividing the polynomial \(f(x)\) by the binomial \((x + 1)\), since \(k = -1\).

Step 4 ：The quotient \(q(x)\) and the remainder \(r\) obtained from the division of \(f(x)\) by \((x + 1)\) are given by \(q(x) = 5x^2 - 4x + 1\) and \(r = -8\).

Step 5 ：\(\boxed{f(x) = (x + 1)(5x^2 - 4x + 1) - 8}\) is the final expression of the polynomial \(f(x)\) in the form \(f(x) = (x - k)q(x) + r\).