Solve (lambda+u)(X+5)=(X-3)(X+2)

$$( \lambda + u ) ( X + 5 ) = ( X - 3 ) ( X + 2 )$$

Solve for u

$$u=-\frac{6+5\lambda +X+X\lambda -X^{2}}{X+5}$$

$$X\neq -5$$

Steps for Solving Linear Equation

Use the distributive property to multiply \lambda +u by X+5.

$$\lambda X+5\lambda +uX+5u=\left(X-3\right)\left(X+2\right)$$

Use the distributive property to multiply X-3 by X+2 and combine like terms.

$$\lambda X+5\lambda +uX+5u=X^{2}-X-6$$

Subtract \lambda X from both sides.

$$5\lambda +uX+5u=X^{2}-X-6-\lambda X$$

Subtract 5\lambda from both sides.

$$uX+5u=X^{2}-X-6-\lambda X-5\lambda$$

Combine all terms containing u.

$$\left(X+5\right)u=X^{2}-X-6-\lambda X-5\lambda$$

The equation is in standard form.

$$\left(X+5\right)u=X^{2}-X\lambda -X-5\lambda -6$$

Divide both sides by X+5.

$$\frac{\left(X+5\right)u}{X+5}=\frac{X^{2}-X\lambda -X-5\lambda -6}{X+5}$$

Dividing by X+5 undoes the multiplication by X+5.

$$u=\frac{X^{2}-X\lambda -X-5\lambda -6}{X+5}$$

Solve for X

$$X=\frac{-\sqrt{u^{2}+2u\lambda +22u+\lambda ^{2}+22\lambda +25}+u+\lambda +1}{2}$$

$$X=\frac{\sqrt{u^{2}+2u\lambda +22u+\lambda ^{2}+22\lambda +25}+u+\lambda +1}{2}\text{, }\lambda \leq -u-4\sqrt{6}-11\text{ or }\lambda \geq -u+4\sqrt{6}-11$$