We have :

\(\left(x+1\right)^2=x^2+2x+1=x^2-529+2x+530\)

\(=\left(x-23\right)\left(x+23\right)+2\left(x+23\right)+484\)

\(=\left(x-21\right)\left(x+23\right)+484\)

Since \(\left(x+1\right)^2⋮x+23\), \(484⋮x+23\)

So, the largest value of x + 23 is 484 and the answer is 461

Call : \(\dfrac{\left(n+1\right)^2}{n+23}=Q_{\left(n\right)}+\dfrac{R}{n+23}\)

n + 23 = 0 => n = -23.

\(R=\left(-23+1\right)^2=\left(-22\right)^2=484\)

\(\Rightarrow\dfrac{\left(n+1\right)^2}{n+23}=Q_{\left(n\right)}+\dfrac{484}{n+3}\in Z\)

\(\Rightarrow n+23\inƯ_{\left(484\right)}=\left\{\pm1;\pm2;\pm4;\pm11;\pm22;\mp44;\pm121;\pm242;\pm484\right\}\)

\(\Rightarrow n+23_{MAX}=484\Rightarrow n=484-23=461\)

Answer : 461.