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express $ x^4 + 3x^3 +4x^2 -6x -12 $ as a product of three factors

i can't do it by means of synthetic division the factors are probably not integers but how else am i supposed to simplify it?

i input it in a calculator and got at most 2 factors $ (x^2 -2)(x^2 +3x +6) $ i know a quadratic expression can have two answers but i can reach this stage by normal means i wanna know what i have to do.

Supporter13
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  • Since you are considering the possibility that the answer won't be all integers, can you factor $x^2-2$ as a difference of squares? – Ben Blum-Smith Nov 02 '13 at 03:20
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    then it will be $ (x- \sqrt2)(x+ \sqrt2)(x^2 +3x +6) $ but i don't know how to get to $ (x^2 -2)(x^2 +3x +6) $ – Supporter13 Nov 02 '13 at 03:23
  • ? You just answered your own question. – Euler....IS_ALIVE Nov 02 '13 at 03:43
  • The general solution to a 4th degree polynomial is a very involved process. I don't know if this one has a trick. In general is it possible to find a solution to a biquadratic just like you would find it for a quadratic, but it's not a formula that's easy to memorize. – DanielV Nov 02 '13 at 03:44
  • first, i didn't answered my own question i just stated again what i was looking for and that is, the means by which this $ x^4 + 3x^3 +4x^2 -6x -12 $ becomes this $ (x^2 -2)(x^2 +3x +6) $ from there, i can express the polynomial as three factors but then again i'm missing a step. second i normally would solve a polynomial with synthetic division but im not getting anything through it in this equation – Supporter13 Nov 02 '13 at 03:52
  • @Euler....IS_ALIVE: The OP is looking for a way to discover that factorization without using a calculator, I believe. – Cameron Buie Nov 02 '13 at 03:53
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    One method is to hope for a factorization of the form $(x^2+ax+b)(x^2+cx+d)$. Expand this out, equate coefficients with the given polynomial, and try to solve the resulting system for $a$, $b$, $c$, and $d$ using substitution. If you run into a nasty polynomial to solve, try the rational root test. – Antonio Vargas Nov 02 '13 at 04:34

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