Solve 2x^2+5x+a+3b/x+2-x-2=8a^2-31b

$$\frac { 2 x ^ { 2 } + 5 x + a + 3 b } { x + 2 } - x - 2 = 8 a ^ { 2 } - 31 b$$

Solve for b (complex solution)

$$\left\{\begin{matrix}b=-\frac{x^{2}-8xa^{2}+x-16a^{2}+a-4}{31x+65}\text{, }&x\neq -\frac{65}{31}\text{ and }x\neq -2\\b\in \mathrm{C}\text{, }&\left(x=-\frac{\sqrt{\frac{122479\sqrt{5786305}}{321408}+\frac{10280872273}{9963648}}}{2}+\frac{\sqrt{5786305}}{288}+\frac{103759}{8928}\text{ and }a=-\frac{\sqrt{5786305}}{1488}-\frac{31}{48}\right)\text{ or }\left(x=-\frac{\sqrt{-\frac{122479\sqrt{5786305}}{321408}+\frac{10280872273}{9963648}}}{2}-\frac{\sqrt{5786305}}{288}+\frac{103759}{8928}\text{ and }a=\frac{\sqrt{5786305}}{1488}-\frac{31}{48}\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by x+2.

$$2x^{2}+5x+a+3b-x\left(x+2\right)+\left(x+2\right)\left(-2\right)=8a^{2}\left(x+2\right)-31b\left(x+2\right)$$

Use the distributive property to multiply -x by x+2.

$$2x^{2}+5x+a+3b-x^{2}-2x+\left(x+2\right)\left(-2\right)=8a^{2}\left(x+2\right)-31b\left(x+2\right)$$

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

$$x^{2}+5x+a+3b-2x+\left(x+2\right)\left(-2\right)=8a^{2}\left(x+2\right)-31b\left(x+2\right)$$

Combine 5x and -2x to get 3x.

$$x^{2}+3x+a+3b+\left(x+2\right)\left(-2\right)=8a^{2}\left(x+2\right)-31b\left(x+2\right)$$

Use the distributive property to multiply x+2 by -2.

$$x^{2}+3x+a+3b-2x-4=8a^{2}\left(x+2\right)-31b\left(x+2\right)$$

Combine 3x and -2x to get x.

$$x^{2}+x+a+3b-4=8a^{2}\left(x+2\right)-31b\left(x+2\right)$$

Use the distributive property to multiply 8a^{2} by x+2.

$$x^{2}+x+a+3b-4=8a^{2}x+16a^{2}-31b\left(x+2\right)$$

Use the distributive property to multiply -31b by x+2.

$$x^{2}+x+a+3b-4=8a^{2}x+16a^{2}-31bx-62b$$

Add 31bx to both sides.

$$x^{2}+x+a+3b-4+31bx=8a^{2}x+16a^{2}-62b$$

Add 62b to both sides.

$$x^{2}+x+a+3b-4+31bx+62b=8a^{2}x+16a^{2}$$

Combine 3b and 62b to get 65b.

$$x^{2}+x+a+65b-4+31bx=8a^{2}x+16a^{2}$$

Subtract x^{2} from both sides.

$$x+a+65b-4+31bx=8a^{2}x+16a^{2}-x^{2}$$

Subtract x from both sides.

$$a+65b-4+31bx=8a^{2}x+16a^{2}-x^{2}-x$$

Subtract a from both sides.

$$65b-4+31bx=8a^{2}x+16a^{2}-x^{2}-x-a$$

Add 4 to both sides.

$$65b+31bx=8a^{2}x+16a^{2}-x^{2}-x-a+4$$

Combine all terms containing b.

$$\left(65+31x\right)b=8a^{2}x+16a^{2}-x^{2}-x-a+4$$

The equation is in standard form.

$$\left(31x+65\right)b=4-a+16a^{2}-x+8xa^{2}-x^{2}$$

Divide both sides by 65+31x.

$$\frac{\left(31x+65\right)b}{31x+65}=\frac{4-a+16a^{2}-x+8xa^{2}-x^{2}}{31x+65}$$

Dividing by 65+31x undoes the multiplication by 65+31x.

$$b=\frac{4-a+16a^{2}-x+8xa^{2}-x^{2}}{31x+65}$$

Solve for b

$$\left\{\begin{matrix}b=-\frac{x^{2}-8xa^{2}+x-16a^{2}+a-4}{31x+65}\text{, }&x\neq -\frac{65}{31}\text{ and }x\neq -2\\b\in \mathrm{R}\text{, }&\left(a=-\frac{\sqrt{5786305}}{1488}-\frac{31}{48}\text{ and }x=-\frac{\sqrt{\frac{122479\sqrt{5786305}}{321408}+\frac{10280872273}{9963648}}}{2}+\frac{\sqrt{5786305}}{288}+\frac{103759}{8928}\right)\text{ or }\left(a=\frac{\sqrt{5786305}}{1488}-\frac{31}{48}\text{ and }x=-\frac{\sqrt{-\frac{122479\sqrt{5786305}}{321408}+\frac{10280872273}{9963648}}}{2}-\frac{\sqrt{5786305}}{288}+\frac{103759}{8928}\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by x+2.

$$2x^{2}+5x+a+3b-x\left(x+2\right)+\left(x+2\right)\left(-2\right)=8a^{2}\left(x+2\right)-31b\left(x+2\right)$$

Use the distributive property to multiply -x by x+2.

$$2x^{2}+5x+a+3b-x^{2}-2x+\left(x+2\right)\left(-2\right)=8a^{2}\left(x+2\right)-31b\left(x+2\right)$$

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

$$x^{2}+5x+a+3b-2x+\left(x+2\right)\left(-2\right)=8a^{2}\left(x+2\right)-31b\left(x+2\right)$$

Combine 5x and -2x to get 3x.

$$x^{2}+3x+a+3b+\left(x+2\right)\left(-2\right)=8a^{2}\left(x+2\right)-31b\left(x+2\right)$$

Use the distributive property to multiply x+2 by -2.

$$x^{2}+3x+a+3b-2x-4=8a^{2}\left(x+2\right)-31b\left(x+2\right)$$

Combine 3x and -2x to get x.

$$x^{2}+x+a+3b-4=8a^{2}\left(x+2\right)-31b\left(x+2\right)$$

Use the distributive property to multiply 8a^{2} by x+2.

$$x^{2}+x+a+3b-4=8a^{2}x+16a^{2}-31b\left(x+2\right)$$

Use the distributive property to multiply -31b by x+2.

$$x^{2}+x+a+3b-4=8a^{2}x+16a^{2}-31bx-62b$$

Add 31bx to both sides.

$$x^{2}+x+a+3b-4+31bx=8a^{2}x+16a^{2}-62b$$

Add 62b to both sides.

$$x^{2}+x+a+3b-4+31bx+62b=8a^{2}x+16a^{2}$$

Combine 3b and 62b to get 65b.

$$x^{2}+x+a+65b-4+31bx=8a^{2}x+16a^{2}$$

Subtract x^{2} from both sides.

$$x+a+65b-4+31bx=8a^{2}x+16a^{2}-x^{2}$$

Subtract x from both sides.

$$a+65b-4+31bx=8a^{2}x+16a^{2}-x^{2}-x$$

Subtract a from both sides.

$$65b-4+31bx=8a^{2}x+16a^{2}-x^{2}-x-a$$

Add 4 to both sides.

$$65b+31bx=8a^{2}x+16a^{2}-x^{2}-x-a+4$$

Combine all terms containing b.

$$\left(65+31x\right)b=8a^{2}x+16a^{2}-x^{2}-x-a+4$$

The equation is in standard form.

$$\left(31x+65\right)b=4-a+16a^{2}-x+8xa^{2}-x^{2}$$

Divide both sides by 65+31x.

$$\frac{\left(31x+65\right)b}{31x+65}=\frac{4-a+16a^{2}-x+8xa^{2}-x^{2}}{31x+65}$$

Dividing by 65+31x undoes the multiplication by 65+31x.

$$b=\frac{4-a+16a^{2}-x+8xa^{2}-x^{2}}{31x+65}$$

Solve for a (complex solution)

$$a=-\frac{\sqrt{32x^{3}+992bx^{2}+96x^{2}+4064bx-64x+4160b-255}-1}{16\left(x+2\right)}$$

$$a=\frac{\sqrt{32x^{3}+992bx^{2}+96x^{2}+4064bx-64x+4160b-255}+1}{16\left(x+2\right)}\text{, }x\neq -2$$

Solve for a

$$a=-\frac{\sqrt{32x^{3}+992bx^{2}+96x^{2}+4064bx-64x+4160b-255}-1}{16\left(x+2\right)}$$

$$a=\frac{\sqrt{32x^{3}+992bx^{2}+96x^{2}+4064bx-64x+4160b-255}+1}{16\left(x+2\right)}\text{, }\left(x\geq -\frac{65}{31}\text{ or }b\geq -\frac{32x^{3}+96x^{2}-64x-255}{992x^{2}+4064x+4160}\right)\text{ and }\left(x>-2\text{ or }x\leq -\frac{65}{31}\text{ or }b\leq -\frac{32x^{3}+96x^{2}-64x-255}{992x^{2}+4064x+4160}\right)\text{ and }\left(x<-2\text{ or }b\geq -\frac{32x^{3}+96x^{2}-64x-255}{992x^{2}+4064x+4160}\right)\text{ and }x\neq -2$$