Answer
$$\left( {x + 2} \right)\sqrt {{{\left( {x + 1} \right)}^3}} $$
Work Step by Step
$$\eqalign{
& \sqrt {{{\left( {x + 1} \right)}^3}} + \sqrt {{{\left( {x + 1} \right)}^5}} \cr
& {\text{Use }}\root n \of {{u^m}} = {u^{m/n}} \cr
& = {\left( {x + 1} \right)^{3/2}} + {\left( {x + 1} \right)^{5/2}} \cr
& {\text{The common factor is }}\left( {x + 1} \right),{\text{ then}} \cr
& {\text{ = }}{\left( {x + 1} \right)^{3/2}}\left[ {\frac{{{{\left( {x + 1} \right)}^{3/2}}}}{{{{\left( {x + 1} \right)}^{3/2}}}} + \frac{{{{\left( {x + 1} \right)}^{5/2}}}}{{{{\left( {x + 1} \right)}^{3/2}}}}} \right] \cr
& {\text{ = }}{\left( {x + 1} \right)^{3/2}}\left[ {1 + {{\left( {x + 1} \right)}^{5/2 - 3/2}}} \right] \cr
& {\text{ = }}{\left( {x + 1} \right)^{3/2}}\left( {1 + x + 1} \right) \cr
& {\text{ = }}\left( {x + 2} \right)\sqrt {{{\left( {x + 1} \right)}^3}} \cr} $$
