Calculate the discriminant of $X^n+aX^{n-1}+b \in \mathbb{Z}[X]$, knowing that the discriminant of $X^n+aX+b$ is $(-1)^{\frac{n^2+n-2}{2}}(n-1)^{n-1}a^n+(-1)^{\frac{n(n-1)}{2}}n^nb^{n-1}$. All I have seen in class is that given a monic polynomial with roots $x_1,...,x_n$ its discriminant is given by $\prod_{i<j}(x_i-x_j)^2$.
I tried first to search a relation between the roots of $X^n+aX^{n-1}+b$ and the roots of $X^n+aX+b$, but trying with small $n$, it seems that there is none.
Then I searched online and I found the definition using the resultant. I tried to write both determinants to see if I could transform one into the other adding rows or columns, but I haven't been able to do so.
Any help please?