Answer
The solutions are $x=-1,0$ and $x=3$
Work Step by Step
$4(x+1)^{1/2}-5(x+1)^{3/2}+(x+1)^{5/2}=0\hspace{0.7cm} \color{blue}{\text{Given equation}}$
$(x+1)^{1/2}\bigg(4-5(x+1)+(x+1)^2\bigg)=0$
$(x+1)^{1/2}\bigg(4-5(x+1)+(x^2+2x+1)\bigg)=0$
$(x+1)^{1/2}\bigg(x^2-3x\bigg)=0$
$(x+1)^{1/2}x(x-3)=0$
$(x+1)^{1/2}=0\hspace{0.5cm}or\hspace{0.5cm}x=0\hspace{0.5cm}or\hspace{0.5cm}x-3=0$
$x=-1\hspace{0.5cm}or\hspace{0.5cm}x=0\hspace{0.5cm}or\hspace{0.5cm}x=3$
The solutions are $x=-1,0$ and $x=3$