One method is guess-and-check. The given polynomial has five terms, so binomial factors won't work (two binomial factors yield four terms, at most).
We could guess trinomial factors.
X^101 + X^52 + X^50 + X^2 - 1 = (? + ? + ?) (? + ? + ?)
The product of the red terms must be -1, yes?
X^101 + X^52 + X^50 + X^2 - 1 = (? + ? + 1) (? + ? - 1)
The product of the blue terms must be X^2. (Can you see why?) Therefore, you know what they are, yes?
The product of the black terms must be X^101. Now it's time to guess and check …