Answer
The maximum number of real zeros is $5$
The number of positive real zeros is $1$
The number of negative real zeros is $0$
Work Step by Step
Number of zeros of a polinomial can’t be greater than its degree
$1)$ Number of positive real zeros of $f(x) $ either equals the number of variations in the sign of the nonzero coefficients of $f(x)$ or equals that number minus an even integer
$2)$ Number of negative real zeros of $f(x)$ either equals the number of variations in the sign of the nonzero coefficients of $f(-x) $ or equals that number minus an even integer.
So the maximum number of real zero here is $5$
Since
$$f\left( x\right) =-3x^{5}+4x^{4}+2 $$ has $1$ variation
The number of positive real zero is $1$
Since
$$f\left( -x\right) =3x^{5}+4x^{4}+2 $$
Has $0$ variations
The number of negative real zeros is $0$