Use FOIL to multiply the two factors:

$\displaystyle{\left({\sqrt[{{4}}]{{{{x}}^{{3}}}}}-{\sqrt[{{4}}]{{{{x}}^{{2}}}}}\right)}{\left({\sqrt[{{4}}]{{{x}}}}+{1}\right)}={\left({\sqrt[{{4}}]{{{{x}}^{{3}}}}}\right)}{\left({\sqrt[{{4}}]{{{x}}}}\right)}+{\left({\sqrt[{{4}}]{{{{x}}^{{3}}}}}\right)}{\left({1}\right)}-{\left({\sqrt[{{4}}]{{{{x}}^{{2}}}}}\right)}{\left({\sqrt[{{4}}]{{{x}}}}\right)}-{\left({\sqrt[{{4}}]{{{{x}}^{{2}}}}}\right)}{\left({1}\right)}\usepackage{cmbright}\Huge\$

$\displaystyle={\sqrt[{{4}}]{{{{x}}^{{4}}}}}+{\sqrt[{{4}}]{{{{x}}^{{3}}}}}-{\sqrt[{{4}}]{{{{x}}^{{3}}}}}-{\sqrt[{{4}}]{{{{x}}^{{2}}}}}={\left|{x}\right|}-{\sqrt[{{4}}]{{{{x}}^{{2}}}}}\usepackage{cmbright}\Huge\$