How to factor $2x^4-11x^3-44x^2+149x+84$

Question

I am doing something for math, and I need to factor $$2x^4-11x^3-44x^2+149x+84.$$ How do we factor it?

Answer 1

I took (another) John's suggestion in the comments and ran through integers between $-10$ and $+10$, to come up with three of the four roots: $-4, 3, 7$.

So the polynomial factors to:

$$(x+4)(x-3)(x-7)(Ax+B)$$

Since we have $2x^4$ in the polynomial, we have $A=2$.

The $x^0$ term is $84$, and the constant terms multiply together to $84$, so $B=1$.

This makes the fourth root $-1/2$ and we have finally

$$2x^4-11x^3-44x^2+149x+84 = (x+4)(x-3)(x-7)(2x+1).$$