Find the values of $\displaystyle a$ such that the equation
$\displaystyle x^4+ax^3+bx^2+ax+1=0$
has $\displaystyle 3$ different real roots.
did you mean "exactly" or "at least" 3 distinct real roots? anyway, the main idea is to note that if $\displaystyle x$ is a root, then $\displaystyle \frac{1}{x}$ will be a root too.
[for those who probably don't know, polynomials of this form can be easily reduced to a quadratic equation by putting $\displaystyle x+\frac{1}{x}=t.$]