Solve y=x^2+px-4/x+r

$$y = \frac { x ^ { 2 } + p x - 4 } { x + r }$$

Solve for p

$$\left\{\begin{matrix}p=\frac{4+ry+xy-x^{2}}{x}\text{, }&x\neq 0\text{ and }x\neq -r\\p\in \mathrm{R}\text{, }&y=-\frac{4}{r}\text{ and }x=0\text{ and }r\neq 0\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by x+r.

$$y\left(x+r\right)=x^{2}+px-4$$

Use the distributive property to multiply y by x+r.

$$yx+yr=x^{2}+px-4$$

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

$$x^{2}+px-4=yx+yr$$

Subtract x^{2} from both sides.

$$px-4=yx+yr-x^{2}$$

Add 4 to both sides.

$$px=yx+yr-x^{2}+4$$

The equation is in standard form.

$$xp=4+ry+xy-x^{2}$$

Divide both sides by x.

$$\frac{xp}{x}=\frac{4+ry+xy-x^{2}}{x}$$

Dividing by x undoes the multiplication by x.

$$p=\frac{4+ry+xy-x^{2}}{x}$$

Divide yx+yr-x^{2}+4 by x.

$$p=\frac{ry+4}{x}+y-x$$

Solve for r

$$\left\{\begin{matrix}r=-\frac{4-px+xy-x^{2}}{y}\text{, }&\left(x=0\text{ or }p\neq -x+\frac{4}{x}\right)\text{ and }y\neq 0\text{ and }x\neq \frac{-\sqrt{p^{2}+16}-p}{2}\text{ and }x\neq \frac{\sqrt{p^{2}+16}-p}{2}\\r\neq -x\text{, }&y=0\text{ and }p=-x+\frac{4}{x}\text{ and }x\neq 0\end{matrix}\right.$$