Which values of $k$ give a maximum at $x=-1$ for $f(x)=(k+1)x^4-(3k+2)x^2-2kx$?

How do you find points of maxima/minima of a given curve?

What is the value of derivative of $f(x)$ at $x=-1$?

How do you determine if a stationary point is a point of maxima?

First derivative of $f(x)$ with respect to $x$ is $(4k+4)x^3-(6k+4)x-2k$, and second derivative of $f(x)$ with respect to $x$ is $(12k+12)x^2-(6k+4)$. Notice that the first derivative is always $0$ at $x=-1,$ it does not depend on $k$. For maxima, the second derivative must be negative at $x=-1$ which gives us $12k+12 -(6k+4)<0,$ and hence $k<-\frac{4}{3}$.