Method of quickly factorizing higher degree polynomials without rational roots?

Question

For a problem I need to factorise $x^4+2x^3-8x^2-6x+15$. The solution is $(x^2+2x-5)(x^2-3)$. Is there some way I can quickly factorise this? This is only a small part of a long and complicated sum which I need to do in 4-5 minutes and I can't spend a lot of time using trial and error method to find the roots, especially when the roots are not rational. Is there some method by which I can factorise this without taking too much time? I cannot use rational root theorem here.

Answer 1

the theorem of Gauss on content says, once there are no rational roots, the remaining possibility is, with integers $a,b,c,d,$ $$ (x^2 + ax+b)(x^2 + cx + d) $$ which can be reduced to four cases, $$ (x^2 + ax+1)(x^2 + cx + 15) \; , $$ $$ (x^2 + ax-1)(x^2 + cx - 15) \; , $$ $$ (x^2 + ax+3)(x^2 + cx + 5) \; , $$ $$ (x^2 + ax-3)(x^2 + cx - 5) \; , $$ so now investigate what $a,c$ might be in each case.

To get the correct $x^3$ $$ (x^2 + ax+1)(x^2 + (2-a)x + 15) \; , $$ $$ (x^2 + ax-1)(x^2 + (2-a)x - 15) \; , $$ $$ (x^2 + ax+3)(x^2 + (2-a)x + 5) \; , $$ $$ (x^2 + ax-3)(x^2 + (2-a)x - 5) \; . $$ For each, you pick $a$ so that the coefficient of $x$ also comes out correct, namely $-6.$ In case $a$ turns out to be an integer, finally check what the coefficient of $x^2$ comes to.