Question

How do you factor `3x^4 - x^3 - 9x^2 + 159x - 52`?

Answer 1

Warning: the following might be (is likely to be) an incomplete solution.

Graphing the `y =` the given expression
That is `y = 3x^4-x^3-9x^2+159x-52`
(see below)
suggests `x=-4` might be a zero for the given expression.

That is `(x+4)` might be a factor of `3x^4-x^3-9x^2+159x-52`

Using synthetic division confirms this possibility and gives
`3x^4-x^3-9x^2+159x-52 = (x+4)(3x^3-13x^2+43x-13)`
as a factorization.

So far I have not been able to factor this second term. Perhaps someone else may do better.
graph{3x^4-x^3-9x^2+159x -52 [-10, 10, -5.21, 5.21]}