Task 5. (20 points) Solve the equation $252 \frac{7}{8}\left(x^{2}+\frac{x^{2}}{(x+1)^{2}}\right)=2023$.
Solution.

$$
\begin{aligned}
& x^{2}+\frac{x^{2}}{(x+1)^{2}}=2023: 252 \frac{7}{8} \\
& x^{2}+\frac{x^{2}}{(x+1)^{2}}=8 \\
& x^{2}-\frac{2 x^{2}}{x+1}+\frac{x^{2}}{(x+1)^{2}}=8-\frac{2 x^{2}}{x+1} \\
& \left(x-\frac{x}{x+1}\right)^{2}=8-\frac{2 x^{2}}{x+1} \\
& \left(\frac{x^{2}}{x+1}\right)^{2}=8-2 \cdot \frac{x^{2}}{x+1}
\end{aligned}
$$

Let's introduce the substitution $\frac{x^{2}}{x+1}=y$.

We get the equation $y^{2}=8-2 y, y^{2}+2 y-8=0 ; y=-4$ or $y=2$

1) if $y=-4$, then $\frac{x^{2}}{x+1}=-4 ; x^{2}+4 x+4=0 ; x=-2$.
2) if $y=2$, then $\frac{x^{2}}{x+1}=2 ; x^{2}-2 x-2=0 ; x=1-\sqrt{3}$ or $x=1+\sqrt{3}$.

Answer. $-2 ; 1-\sqrt{3} ; 1+\sqrt{3}$.