Example: solve the equation \(x\left( {x + 1} \right)({x^2} + x + 1) = 42\)

Solution: Suppose \(x(x + 1) = t\), then \({x^2} + x = t\) or  

\({x^2} + x + {\rm{ }}1{\rm{ }} = t + {\rm{ }}1\), hence 

\(t(t + 1) = 42\) or \(t = 6\) or -7

Case 1: If \(t = 6  \Rightarrow  x(x + 1)=6\) or \(x = 2\) or -3

Case 2: If \(t = - 7\) \( \Rightarrow \)  \((x) (x + 1) = -7\) 

 Or \({x^2} + x + {\rm{ }}7{\rm{ }} = {\rm{ }}0\) \( \Rightarrow \) x =\(\frac{{ - 1 \pm \sqrt { - 27} }}{2} = \frac{{ - 1 \pm \sqrt {27} \,\,i}}{2}\)