Solve n*(n-1)!+(n-1)!=x*(n+1)*n*(n-1)!

$$n \cdot ( n - 1 ) ! + ( n - 1 ) ! = x \cdot ( n + 1 ) \cdot n \cdot ( n - 1 ) !$$

Solve for x (complex solution)

$$\left\{\begin{matrix}x=\frac{1}{n}\text{, }&n\neq 0\\x\in \mathrm{C}\text{, }&\left(n-1\right)!=0\text{ or }n=-1\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply x by n+1.

$$n\left(n-1\right)!+\left(n-1\right)!=\left(xn+x\right)n\left(n-1\right)!$$

Use the distributive property to multiply xn+x by n.

$$n\left(n-1\right)!+\left(n-1\right)!=\left(xn^{2}+xn\right)\left(n-1\right)!$$

Use the distributive property to multiply xn^{2}+xn by \left(n-1\right)!.

$$n\left(n-1\right)!+\left(n-1\right)!=xn^{2}\left(n-1\right)!+xn\left(n-1\right)!$$

Swap sides so that all variable terms are on the left hand side.

$$xn^{2}\left(n-1\right)!+xn\left(n-1\right)!=n\left(n-1\right)!+\left(n-1\right)!$$

Combine all terms containing x.

$$\left(n^{2}\left(n-1\right)!+n\left(n-1\right)!\right)x=n\left(n-1\right)!+\left(n-1\right)!$$

Divide both sides by n^{2}\left(n-1\right)!+n\left(n-1\right)!.

$$\frac{\left(n^{2}\left(n-1\right)!+n\left(n-1\right)!\right)x}{n^{2}\left(n-1\right)!+n\left(n-1\right)!}=\frac{\left(n+1\right)\left(n-1\right)!}{n^{2}\left(n-1\right)!+n\left(n-1\right)!}$$

Dividing by n^{2}\left(n-1\right)!+n\left(n-1\right)! undoes the multiplication by n^{2}\left(n-1\right)!+n\left(n-1\right)!.

$$x=\frac{\left(n+1\right)\left(n-1\right)!}{n^{2}\left(n-1\right)!+n\left(n-1\right)!}$$

Divide \left(n-1\right)!\left(1+n\right) by n^{2}\left(n-1\right)!+n\left(n-1\right)!.

$$x=\frac{1}{n}$$

Solve for x

$$\left\{\begin{matrix}x=\frac{1}{n}\text{, }&n\neq 0\\x\in \mathrm{R}\text{, }&\left(n-1\right)!=0\text{ or }n=-1\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply x by n+1.

$$n\left(n-1\right)!+\left(n-1\right)!=\left(xn+x\right)n\left(n-1\right)!$$

Use the distributive property to multiply xn+x by n.

$$n\left(n-1\right)!+\left(n-1\right)!=\left(xn^{2}+xn\right)\left(n-1\right)!$$

Use the distributive property to multiply xn^{2}+xn by \left(n-1\right)!.

$$n\left(n-1\right)!+\left(n-1\right)!=xn^{2}\left(n-1\right)!+xn\left(n-1\right)!$$

Swap sides so that all variable terms are on the left hand side.

$$xn^{2}\left(n-1\right)!+xn\left(n-1\right)!=n\left(n-1\right)!+\left(n-1\right)!$$

Combine all terms containing x.

$$\left(n^{2}\left(n-1\right)!+n\left(n-1\right)!\right)x=n\left(n-1\right)!+\left(n-1\right)!$$

Divide both sides by n^{2}\left(n-1\right)!+n\left(n-1\right)!.

$$\frac{\left(n^{2}\left(n-1\right)!+n\left(n-1\right)!\right)x}{n^{2}\left(n-1\right)!+n\left(n-1\right)!}=\frac{\left(n+1\right)\left(n-1\right)!}{n^{2}\left(n-1\right)!+n\left(n-1\right)!}$$

Dividing by n^{2}\left(n-1\right)!+n\left(n-1\right)! undoes the multiplication by n^{2}\left(n-1\right)!+n\left(n-1\right)!.

$$x=\frac{\left(n+1\right)\left(n-1\right)!}{n^{2}\left(n-1\right)!+n\left(n-1\right)!}$$

Divide \left(n-1\right)!\left(1+n\right) by n^{2}\left(n-1\right)!+n\left(n-1\right)!.

$$x=\frac{1}{n}$$