how do you factor this polynomial function? 4x^6 + 12x^5-9x^4-51x^3-30x^2+12x+8

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I think there are a couple of ways to factor this but you have to start somewhere.  I always look at x = +1 or -1 to start out.  It turns out that x = -1 is a factor.  Then, you can use synthetic division to determine subsequent factors but remember there can only be a maximum of 6 factors for this 6th degree polynomial.  If you have a graphing calculator just plug this equation in to find the zeroes.  x = -2 is another real factor, the rest is up to you.