Has the polynomial distinct roots? How can I prove it?

I want to prove that the polynomial

$$
f_p(x) = x^{2p+2} – cx^{2p} – dx^p – 1
$$

,where $c>0$ and $d>0$ are real numbers, has distinct roots. Also $p>0$ is an even integer. How can I prove that the polynomial $f_p(x)$ has distinct roots for any $c$,$d$ and $p$.

PS: There is a similar topic that How to prove that my polynomial has distinct roots?

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Here is a partial solution that shows there can be no multiple REAL roots. The proof doesn’t work for complex numbers though (and I am not sure the result is even true in complex numbers).

$f’ = (2p+2) x^{2p+1} – 2pcx^{2p-1} – pdx^{p-1}$. If $x$ is a multiple root of $f$, then both $f$ and $f’$ vanish at $x$. But $(2p+2) f – xf’ = -2cx^{2p}-(p+2)dx^p-(2p+2) < 0$ for all $x \in \mathbb R$ because $p$ is even and $c$, $d$ and $p$ are all positive.

‘All’ you have to do is computing the g.c.d. of $f_p(x)$ and $f’_p(x)$ via the Euclidean algorithm. A polynomial has only simple roots (in $\mathbf C$) if and only if this polynomial and its derivative are coprime.