Solve (m-1)x^2+2m*x+m+3=0

$$( m - 1 ) x ^ { 2 } + 2 m \cdot x + m + 3 = 0$$

Solve for m

$$m=-\frac{3-x^{2}}{\left(x+1\right)^{2}}$$

$$x\neq -1$$

Steps for Solving Linear Equation

Use the distributive property to multiply m-1 by x^{2}.

$$mx^{2}-x^{2}+2mx+m+3=0$$

Add x^{2} to both sides. Anything plus zero gives itself.

$$mx^{2}+2mx+m+3=x^{2}$$

Subtract 3 from both sides.

$$mx^{2}+2mx+m=x^{2}-3$$

Combine all terms containing m.

$$\left(x^{2}+2x+1\right)m=x^{2}-3$$

Divide both sides by x^{2}+2x+1.

$$\frac{\left(x^{2}+2x+1\right)m}{x^{2}+2x+1}=\frac{x^{2}-3}{x^{2}+2x+1}$$

Dividing by x^{2}+2x+1 undoes the multiplication by x^{2}+2x+1.

$$m=\frac{x^{2}-3}{x^{2}+2x+1}$$

Divide x^{2}-3 by x^{2}+2x+1.

$$m=\frac{x^{2}-3}{\left(x+1\right)^{2}}$$

Solve for x (complex solution)

$$\left\{\begin{matrix}x=\frac{-m+\sqrt{3-2m}}{m-1}\text{; }x=-\frac{m+\sqrt{3-2m}}{m-1}\text{, }&m\neq 1\\x=-2\text{, }&m=1\end{matrix}\right.$$

Solve for x

$$\left\{\begin{matrix}x=\frac{-m+\sqrt{3-2m}}{m-1}\text{; }x=-\frac{m+\sqrt{3-2m}}{m-1}\text{, }&m\neq 1\text{ and }m\leq \frac{3}{2}\\x=-2\text{, }&m=1\end{matrix}\right.$$