Solve -:   x(x+1)^2(x+2) = 72

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$$\begin{gathered} \mathrm{We} \mathrm{have} \\ \mathrm{x}{\left(\mathrm{x}+1\right)}^2\left(\mathrm{x}+2\right)=72 \\ \Rightarrow \left(\mathrm{x}+1-1\right){\left(\mathrm{x}+1\right)}^2\left(\mathrm{x}+1+1\right)=72 \\ \mathrm{Let} \mathrm{x}+1=\mathrm{t} \\ \Rightarrow \left(\mathrm{t}-1\right).{\mathrm{t}}^2\left(\mathrm{t}+1\right)=72 \\ \Rightarrow \left(\mathrm{t}-1\right)\left(\mathrm{t}+1\right){\mathrm{t}}^2=72 \\ \Rightarrow \left({\mathrm{t}}^2-1\right).{\mathrm{t}}^2-72=0 \\ \Rightarrow {\mathrm{t}}^4-{\mathrm{t}}^2-72=0 \\ \Rightarrow {\mathrm{t}}^4-9{\mathrm{t}}^2+8{\mathrm{t}}^2-72=0 \\ \Rightarrow {\mathrm{t}}^2\left({\mathrm{t}}^2-9\right)+8\left({\mathrm{t}}^2-9\right)=0 \\ \Rightarrow \left({\mathrm{t}}^2+8\right)\left({\mathrm{t}}^2-9\right)=0 \\ \mathrm{But} {\mathrm{t}}^2+8\ne 0 \\ \Rightarrow {\mathrm{t}}^2-9=0 \\ \Rightarrow {\mathrm{t}}^2=9 \\ \Rightarrow \mathrm{t}=3 \mathrm{or} \mathrm{t}=-3 \\ \Rightarrow \mathrm{x}+1=3 \mathrm{or} \mathrm{x}+1=-3 \\ \mathbf{\Rightarrow }\mathbf{x}\mathbf{=}\mathbf{2}\mathbf{ }\mathbf{or}\mathbf{ }\mathbf{x}\mathbf{=}\mathbf{-}\mathbf{4}\mathbf{ }\left(\mathbf{Answer}\right) \end{gathered}$$

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