$\displaystyle \frac{x-1}{x^{\frac{3}{4}}+x^{\frac{1}{2}}}\cdot\frac{x^{\ frac{1}{2}}+x^{\frac{1}{4}}}{x^{\frac{1}{2}}+1}\cd ot x^{\frac{1}{4}}+1$
How it begins?
$\displaystyle \dfrac{x-1}{x^{\frac{3}{4}}+x^{\frac{1}{2}}}\cdot\dfrac{x^{ \frac{1}{2}}+x^{\frac{1}{4}}}{x^{\frac{1}{2}}+1}\c dot x^{\frac{1}{4}}+1$
Factorise where you can, and you'll see some terms crossing out.
For example:
$\displaystyle x^\frac34 + x^\frac12 = x^\frac34 + x^\frac24 = x^\frac24(x^\frac14 + 1)$