Answer
Maximum number of zeros: $4.$
1 positive, 1 negative zero.
Work Step by Step
$(-x)^{n}=\left\{\begin{array}{ll}
x^{n} & \text{if n is even}\\
-x^{n} & \text{if n is odd}
\end{array}\right.$
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$f(x)=5x^{4}+2x^{2}-6x-5$
has 1 change in signs, from $+2x^{2}$ to $-6x$.
By Descartes’ Rule of Signs, f has 1 positive zero.
$f(-x)=5x^{4}+2x^{2}+6x-5$
has 1 change in signs, so it has 1 negative zero.
The degree of the polynomial is 4, which is the maximum number of zeros.