Factoring 4th degree polynomials.

[cp255]

So I needed to factor -4x⁵-8x⁴+8x³+4x.
I factored out a -4x and I am left with x⁴+2x³-2x²-4.

The problem is I am unsure how to factor x⁴+2x³-2x²-4.
I know how to long divide polynomials but I have not done synthetic division in over 4 years. From what I have seen on the internet it seems like a lot of guess and check.

 

[jbunniii]

You made a mistake when you divided by -4x. After dividing, the last term should be -1, not -4.

You can find a root by trial and error, or by using the rational root theorem: http://en.wikipedia.org/wiki/Rational_root_theorem

 

[epenguin]

Easier, it makes sense to group the pair with factor 2 together - you can certainly factor that pair. The remaining two terms have a factorisation that is very like one you should remember - since you say you are out of practice perhaps the only one you'd remember.

 

[eddybob123]

There is a quartic formula which you can use, and once you know its solutions, you know its factors. The only porblem is that it is extremely not recommended...
http://www.curtisbright.com/quartic/quartic-png.html