How to factor the polynomial $8x^3-16x-85$?

Question

While solving a problem on sequences and series, I got the following cubic equation

$$8x^3-16x-85=0$$

I cannot figure out how to solve it. I have tried to factor the L.H.S., but did not succeed. Please help. Are these type of equations solvable by factorization or is using a particular formula the only way?

Answer 1

Hint. Find a rational zero $p/q$ such that $p$ divides $85$ and $q$ divides $8$. Note also that $f(2)=-53<0$ and $f(3)=83>0$.

Answer 2

Hint: Assume that $8x^3-16x-85=(ax+b)(cx^2+dx+f)$ and compare coefficients. Then we see that $$ 8x^3-16x-85=(2x-5)(4x^2+10x+17). $$