Solve 2x^2-x+q-5=(2x+1)(x+p)

$$2 x ^ { 2 } - x + q - 5 = ( 2 x + 1 ) ( x + p )$$

Solve for p (complex solution)

$$\left\{\begin{matrix}p=-\frac{2x-q+5}{2x+1}\text{, }&x\neq -\frac{1}{2}\\p\in \mathrm{C}\text{, }&q=4\text{ and }x=-\frac{1}{2}\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply 2x+1 by x+p.

$$2x^{2}-x+q-5=2x^{2}+2xp+x+p$$

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

$$2x^{2}+2xp+x+p=2x^{2}-x+q-5$$

Subtract 2x^{2} from both sides.

$$2xp+x+p=2x^{2}-x+q-5-2x^{2}$$

Combine 2x^{2} and -2x^{2} to get 0.

$$2xp+x+p=-x+q-5$$

Subtract x from both sides.

$$2xp+p=-x+q-5-x$$

Combine -x and -x to get -2x.

$$2xp+p=-2x+q-5$$

Combine all terms containing p.

$$\left(2x+1\right)p=-2x+q-5$$

Divide both sides by 2x+1.

$$\frac{\left(2x+1\right)p}{2x+1}=\frac{-2x+q-5}{2x+1}$$

Dividing by 2x+1 undoes the multiplication by 2x+1.

$$p=\frac{-2x+q-5}{2x+1}$$

Solve for p

$$\left\{\begin{matrix}p=-\frac{2x-q+5}{2x+1}\text{, }&x\neq -\frac{1}{2}\\p\in \mathrm{R}\text{, }&q=4\text{ and }x=-\frac{1}{2}\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply 2x+1 by x+p.

$$2x^{2}-x+q-5=2x^{2}+2xp+x+p$$

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

$$2x^{2}+2xp+x+p=2x^{2}-x+q-5$$

Subtract 2x^{2} from both sides.

$$2xp+x+p=2x^{2}-x+q-5-2x^{2}$$

Combine 2x^{2} and -2x^{2} to get 0.

$$2xp+x+p=-x+q-5$$

Subtract x from both sides.

$$2xp+p=-x+q-5-x$$

Combine -x and -x to get -2x.

$$2xp+p=-2x+q-5$$

Combine all terms containing p.

$$\left(2x+1\right)p=-2x+q-5$$

Divide both sides by 2x+1.

$$\frac{\left(2x+1\right)p}{2x+1}=\frac{-2x+q-5}{2x+1}$$

Dividing by 2x+1 undoes the multiplication by 2x+1.

$$p=\frac{-2x+q-5}{2x+1}$$

Solve for q

$$q=2px+2x+p+5$$

Solution Steps

Use the distributive property to multiply 2x+1 by x+p.

$$2x^{2}-x+q-5=2x^{2}+2xp+x+p$$

Subtract 2x^{2} from both sides.

$$-x+q-5=2x^{2}+2xp+x+p-2x^{2}$$

Combine 2x^{2} and -2x^{2} to get 0.

$$-x+q-5=2xp+x+p$$

Add x to both sides.

$$q-5=2xp+x+p+x$$

Combine x and x to get 2x.

$$q-5=2xp+2x+p$$

Add 5 to both sides.

$$q=2xp+2x+p+5$$