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

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