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.
...perform long division by (x-2)(x+3).
I hope this helps. Good luck!!