Find the value(s) of $k$ for which the function $f(x)=\frac{x^5}{5}+\frac{x^4}{4}+x^3+\frac{k}{2}x^2+x$ is increasing for all $x\in \mathbb{R}$.

Question

Find the value(s) of $k$ for which the function $$f(x)=\frac{x^5}{5}+\frac{x^4}{4}+x^3+\frac{k}{2}x^2+x$$ is always increasing for all $x\in \mathbb{R}$.

My Attempt

$f'(x)=x^4+x^3+3x^2+1+kx=\left(x^2+\frac{x}{2}+1\right)^2+\frac{3}{4}x^2+(k-1)x$

But I am not able to comprehend after this