Solve f(x)=4x+3/x^2+1(xinR)

$$f ( x ) = \frac { 4 x + 3 } { x ^ { 2 } + 1 } ( x \in R )$$

Solve for f (complex solution)

$$\left\{\begin{matrix}f=\frac{\left(4x+3\right)\left(x\in R\right)}{x\left(x^{2}+1\right)}\text{, }&x\neq -i\text{ and }x\neq i\text{ and }x\neq 0\\f\in \mathrm{C}\text{, }&\left(0\in R\right)=0\text{ and }x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by \left(x-i\right)\left(x+i\right).

$$fx\left(x-i\right)\left(x+i\right)=\left(4x+3\right)\left(x\in R\right)$$

Use the distributive property to multiply fx by x-i.

$$\left(fx^{2}-ifx\right)\left(x+i\right)=\left(4x+3\right)\left(x\in R\right)$$

Use the distributive property to multiply fx^{2}-ifx by x+i and combine like terms.

$$fx^{3}+xf=\left(4x+3\right)\left(x\in R\right)$$

Use the distributive property to multiply 4x+3 by x\in R.

$$fx^{3}+xf=4x\left(x\in R\right)+3\left(x\in R\right)$$

Combine all terms containing f.

$$\left(x^{3}+x\right)f=4x\left(x\in R\right)+3\left(x\in R\right)$$

Divide both sides by x^{3}+x.

$$\frac{\left(x^{3}+x\right)f}{x^{3}+x}=\frac{\left(4x+3\right)\left(x\in R\right)}{x^{3}+x}$$

Dividing by x^{3}+x undoes the multiplication by x^{3}+x.

$$f=\frac{\left(4x+3\right)\left(x\in R\right)}{x^{3}+x}$$

Divide \left(3+4x\right)\left(x\in R\right) by x^{3}+x.

$$f=\frac{\left(4x+3\right)\left(x\in R\right)}{x\left(x^{2}+1\right)}$$

Solve for f

$$\left\{\begin{matrix}f=\frac{\left(4x+3\right)\left(x\in R\right)}{x\left(x^{2}+1\right)}\text{, }&x\neq 0\\f\in \mathrm{R}\text{, }&\left(0\in R\right)=0\text{ and }x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by x^{2}+1.

$$fx\left(x^{2}+1\right)=\left(4x+3\right)\left(x\in R\right)$$

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

$$fx^{3}+fx=\left(4x+3\right)\left(x\in R\right)$$

Use the distributive property to multiply 4x+3 by x\in R.

$$fx^{3}+fx=4x\left(x\in R\right)+3\left(x\in R\right)$$

Combine all terms containing f.

$$\left(x^{3}+x\right)f=4x\left(x\in R\right)+3\left(x\in R\right)$$

Divide both sides by x^{3}+x.

$$\frac{\left(x^{3}+x\right)f}{x^{3}+x}=\frac{\left(4x+3\right)\left(x\in R\right)}{x^{3}+x}$$

Dividing by x^{3}+x undoes the multiplication by x^{3}+x.

$$f=\frac{\left(4x+3\right)\left(x\in R\right)}{x^{3}+x}$$

Divide \left(3+4x\right)\left(x\in R\right) by x^{3}+x.

$$f=\frac{\left(4x+3\right)\left(x\in R\right)}{x\left(x^{2}+1\right)}$$