how do you factor 3x^-1/2 + 4x^1/2 + x^3/2?
It might help to substitute u for √x. That gets rid of all those pesky fractional exponents.
3/u + 4u + u^3
(3+4u+u^4)/u
(u+1)^2 (u^2-2u+3)/u
(√x+1)^2 (x^3/2 - 2√x + 3/√x)
The hard part here is factoring u^4+4u+3.
Try synthetic division, knowing that any rational roots must be ±1 or ±3. Luckily, -1 pops up twice.