Solution.

From the condition, we have $\left(x^{2}+2 x\right)^{2}-\left(x^{2}+2 x+1\right)=55$. Let $x^{2}+2 x=y$.

The equation in terms of $y$ becomes $y^{2}-y-56=0 \Rightarrow y_{1}=-7, y_{2}=8$.

Then $x^{2}+2 x=-7$ or $x^{2}+2 x=8, x^{2}+2 x+7=0$ or $x^{2}+2 x-8=0$, from which $x_{1}=-4, x_{2}=2$. The first equation has no solutions, since $D<0$.

Answer: $x_{1}=-4, x_{2}=2$.
6.144. $\left(x^{2}+2 x\right)^{2}-(x+1)^{2}=55$.

6.144. $\left(x^{2}+2 x\right)^{2}-(x+1)^{2}=55$.