How to find all the solutions to $-8x^4-18x^2-11+\frac{1}{x^4}=0$? How to know all the roots for fourth degree  equation like this :
$$-8x^4-18x^2-11+\frac{1}{x^4}=0$$

Rewrite as $8x^8+18x^6+11x^4-1=0$. By inspection, $i$ is a root, and therefore so is $-i$. Divide the polynomial by $x^2+1$. We get $8x^6+10x^4+x^2-1$. Note that $i$ is a root of this, and therefore so is $-i$. Divide again by $x^2+1$. We get $8x^4+2x^2-1$. Let $y=x^2$. Now we have an easy quadratic.
Remark: Please note the comment by lab bhattacharjee. Things are easier still if we let $x^2=y$ immediately. Then we are looking at $8y^4+18y^3+11y^2-1=0$, and the root $-1$ is obvious.  Divide by $y+1$, and continue. 
