Solve (k+1)x^2+(k+3)x+k=0

$$( k + 1 ) x ^ { 2 } + ( k + 3 ) x + k = 0$$

Solve for k (complex solution)

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

$$x\neq \frac{-1+\sqrt{3}i}{2}\text{ and }x\neq \frac{-\sqrt{3}i-1}{2}$$

Steps for Solving Linear Equation

Use the distributive property to multiply k+1 by x^{2}.

$$kx^{2}+x^{2}+\left(k+3\right)x+k=0$$

Use the distributive property to multiply k+3 by x.

$$kx^{2}+x^{2}+kx+3x+k=0$$

Subtract x^{2} from both sides. Anything subtracted from zero gives its negation.

$$kx^{2}+kx+3x+k=-x^{2}$$

Subtract 3x from both sides.

$$kx^{2}+kx+k=-x^{2}-3x$$

Combine all terms containing k.

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

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

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

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

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

Solve for k

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

Steps for Solving Linear Equation

Use the distributive property to multiply k+1 by x^{2}.

$$kx^{2}+x^{2}+\left(k+3\right)x+k=0$$

Use the distributive property to multiply k+3 by x.

$$kx^{2}+x^{2}+kx+3x+k=0$$

Subtract x^{2} from both sides. Anything subtracted from zero gives its negation.

$$kx^{2}+kx+3x+k=-x^{2}$$

Subtract 3x from both sides.

$$kx^{2}+kx+k=-x^{2}-3x$$

Combine all terms containing k.

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

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

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

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

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

Solve for x (complex solution)

$$\left\{\begin{matrix}x=\frac{\sqrt{9+2k-3k^{2}}-k-3}{2\left(k+1\right)}\text{; }x=-\frac{\sqrt{9+2k-3k^{2}}+k+3}{2\left(k+1\right)}\text{, }&k\neq -1\\x=\frac{1}{2}\text{, }&k=-1\end{matrix}\right.$$

Solve for x

$$\left\{\begin{matrix}x=\frac{\sqrt{9+2k-3k^{2}}-k-3}{2\left(k+1\right)}\text{; }x=-\frac{\sqrt{9+2k-3k^{2}}+k+3}{2\left(k+1\right)}\text{, }&k\neq -1\text{ and }k\geq \frac{1-2\sqrt{7}}{3}\text{ and }k\leq \frac{2\sqrt{7}+1}{3}\\x=\frac{1}{2}\text{, }&k=-1\end{matrix}\right.$$