Try plugging small numbers like -3, -2, -1, 1, 2, 3. If any of them is a zero, then one of the factors is known.

For your examples, check to see that x=-3 and x=2 are zeros, which means (x-2) and (x+3) are factors.

You can either do synthetic division of 6x^4-27x^3-54x^2+213x-90 by (x-2).

Then take the quotient and divide it synthetically by x+3.

The quotient at this stage will be a quadratic function, which should be easy to factor.

OR

...perform long division by (x-2)(x+3).

I hope this helps. Good luck!!

Spoiler: http://www.wolframalpha.com/input/?i=6x^4-27x^3-54x^2%2B213x-90%3D0+solve