Solve (3x+1)(x-1)-(x+3)(5x-6)=x^2-10x+m

$$( 3 x + 1 ) ( x - 1 ) - ( x + 3 ) ( 5 x - 6 ) = x ^ { 2 } - 10 x + m$$

Solve for m

$$m=17-x-3x^{2}$$

Solution Steps

Use the distributive property to multiply 3x+1 by x-1 and combine like terms.

$$3x^{2}-2x-1-\left(x+3\right)\left(5x-6\right)=x^{2}-10x+m$$

Use the distributive property to multiply x+3 by 5x-6 and combine like terms.

$$3x^{2}-2x-1-\left(5x^{2}+9x-18\right)=x^{2}-10x+m$$

To find the opposite of 5x^{2}+9x-18, find the opposite of each term.

$$3x^{2}-2x-1-5x^{2}-9x+18=x^{2}-10x+m$$

Combine 3x^{2} and -5x^{2} to get -2x^{2}.

$$-2x^{2}-2x-1-9x+18=x^{2}-10x+m$$

Combine -2x and -9x to get -11x.

$$-2x^{2}-11x-1+18=x^{2}-10x+m$$

Add -1 and 18 to get 17.

$$-2x^{2}-11x+17=x^{2}-10x+m$$

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

$$x^{2}-10x+m=-2x^{2}-11x+17$$

Subtract x^{2} from both sides.

$$-10x+m=-2x^{2}-11x+17-x^{2}$$

Combine -2x^{2} and -x^{2} to get -3x^{2}.

$$-10x+m=-3x^{2}-11x+17$$

Add 10x to both sides.

$$m=-3x^{2}-11x+17+10x$$

Combine -11x and 10x to get -x.

$$m=-3x^{2}-x+17$$

Solve for x (complex solution)

$$x=\frac{-\sqrt{205-12m}-1}{6}$$

$$x=\frac{\sqrt{205-12m}-1}{6}$$

Solve for x

$$x=\frac{-\sqrt{205-12m}-1}{6}$$

$$x=\frac{\sqrt{205-12m}-1}{6}\text{, }m\leq \frac{205}{12}$$