Answer
See the explanation
Work Step by Step
By Descartes' rule of signs, if a polynomial in one variable, $f(x) = a_n x^n + a_{n-1}x^{n-1} + a_{n-2}x^{n-2} + ...+ a_{1}x + a_{0}$ is arranged in the descending order of the exponents of the variable, then:
The number of positive real zeros of $f(x)$ is either equal to the number of sign changes in $f(x)$ or less than the number of sign changes by an even number.
The same rule applies to find the number of negative real zeros as well, but then we count the sign changes of $f(-x)$.
Thus, $P(x)=3x^4+5x^2+2$, has no sign change and, therefore has no positive roots.
$P(-x)=3x^4+5x^2+2$, has no sign change as well, therefore $P(x)$ has no negative roots.
We conclude that the polynomial has no real roots.
