Solve {l}{8x+16+8/x+2*7/x-2=8x^2-25/x-2}{8(x+2)+}

$$\left. \begin{array} { l } { 8 x + 16 + \frac { 8 } { x + 2 } \cdot \frac { 7 } { x - 2 } = \frac { 8 x ^ { 2 } - 25 } { x - 2 } } \\ { 8 ( x + 2 ) + } \end{array} \right.$$

Solve for x

$$x=6$$

Steps Using Factoring By Grouping

Variable x cannot be equal to any of the values -2,2 since division by zero is not defined. Multiply both sides of the equation by \left(x-2\right)\left(x+2\right), the least common multiple of x+2,x-2.

$$8x\left(x-2\right)\left(x+2\right)+\left(x-2\right)\left(x+2\right)\times 16+\left(x-2\right)\times 8\times \frac{7}{x-2}=\left(x+2\right)\left(8x^{2}-25\right)$$

Use the distributive property to multiply 8x by x-2.

$$\left(8x^{2}-16x\right)\left(x+2\right)+\left(x-2\right)\left(x+2\right)\times 16+\left(x-2\right)\times 8\times \frac{7}{x-2}=\left(x+2\right)\left(8x^{2}-25\right)$$

Use the distributive property to multiply 8x^{2}-16x by x+2 and combine like terms.

$$8x^{3}-32x+\left(x-2\right)\left(x+2\right)\times 16+\left(x-2\right)\times 8\times \frac{7}{x-2}=\left(x+2\right)\left(8x^{2}-25\right)$$

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

$$8x^{3}-32x+\left(x^{2}-4\right)\times 16+\left(x-2\right)\times 8\times \frac{7}{x-2}=\left(x+2\right)\left(8x^{2}-25\right)$$

Use the distributive property to multiply x^{2}-4 by 16.

$$8x^{3}-32x+16x^{2}-64+\left(x-2\right)\times 8\times \frac{7}{x-2}=\left(x+2\right)\left(8x^{2}-25\right)$$

Express \left(x-2\right)\times \frac{7}{x-2} as a single fraction.

$$8x^{3}-32x+16x^{2}-64+\frac{\left(x-2\right)\times 7}{x-2}\times 8=\left(x+2\right)\left(8x^{2}-25\right)$$

Use the distributive property to multiply x+2 by 8x^{2}-25.

$$8x^{3}-32x+16x^{2}-64+\frac{\left(x-2\right)\times 7}{x-2}\times 8=8x^{3}-25x+16x^{2}-50$$

Use the distributive property to multiply x-2 by 7.

$$8x^{3}-32x+16x^{2}-64+\frac{7x-14}{x-2}\times 8=8x^{3}-25x+16x^{2}-50$$

Express \frac{7x-14}{x-2}\times 8 as a single fraction.

$$8x^{3}-32x+16x^{2}-64+\frac{\left(7x-14\right)\times 8}{x-2}=8x^{3}-25x+16x^{2}-50$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 8x^{3}-32x+16x^{2}-64 times \frac{x-2}{x-2}.

$$\frac{\left(8x^{3}-32x+16x^{2}-64\right)\left(x-2\right)}{x-2}+\frac{\left(7x-14\right)\times 8}{x-2}=8x^{3}-25x+16x^{2}-50$$

Since \frac{\left(8x^{3}-32x+16x^{2}-64\right)\left(x-2\right)}{x-2} and \frac{\left(7x-14\right)\times 8}{x-2} have the same denominator, add them by adding their numerators.

$$\frac{\left(8x^{3}-32x+16x^{2}-64\right)\left(x-2\right)+\left(7x-14\right)\times 8}{x-2}=8x^{3}-25x+16x^{2}-50$$

Do the multiplications in \left(8x^{3}-32x+16x^{2}-64\right)\left(x-2\right)+\left(7x-14\right)\times 8.

$$\frac{8x^{4}-16x^{3}-32x^{2}+64x+16x^{3}-32x^{2}-64x+128+56x-112}{x-2}=8x^{3}-25x+16x^{2}-50$$

Combine like terms in 8x^{4}-16x^{3}-32x^{2}+64x+16x^{3}-32x^{2}-64x+128+56x-112.

$$\frac{8x^{4}-64x^{2}+56x+16}{x-2}=8x^{3}-25x+16x^{2}-50$$

Subtract 8x^{3} from both sides.

$$\frac{8x^{4}-64x^{2}+56x+16}{x-2}-8x^{3}=-25x+16x^{2}-50$$

To add or subtract expressions, expand them to make their denominators the same. Multiply -8x^{3} times \frac{x-2}{x-2}.

$$\frac{8x^{4}-64x^{2}+56x+16}{x-2}+\frac{-8x^{3}\left(x-2\right)}{x-2}=-25x+16x^{2}-50$$

Since \frac{8x^{4}-64x^{2}+56x+16}{x-2} and \frac{-8x^{3}\left(x-2\right)}{x-2} have the same denominator, add them by adding their numerators.

$$\frac{8x^{4}-64x^{2}+56x+16-8x^{3}\left(x-2\right)}{x-2}=-25x+16x^{2}-50$$

Do the multiplications in 8x^{4}-64x^{2}+56x+16-8x^{3}\left(x-2\right).

$$\frac{8x^{4}-64x^{2}+56x+16-8x^{4}+16x^{3}}{x-2}=-25x+16x^{2}-50$$

Combine like terms in 8x^{4}-64x^{2}+56x+16-8x^{4}+16x^{3}.

$$\frac{-64x^{2}+56x+16+16x^{3}}{x-2}=-25x+16x^{2}-50$$

Add 25x to both sides.

$$\frac{-64x^{2}+56x+16+16x^{3}}{x-2}+25x=16x^{2}-50$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 25x times \frac{x-2}{x-2}.

$$\frac{-64x^{2}+56x+16+16x^{3}}{x-2}+\frac{25x\left(x-2\right)}{x-2}=16x^{2}-50$$

Since \frac{-64x^{2}+56x+16+16x^{3}}{x-2} and \frac{25x\left(x-2\right)}{x-2} have the same denominator, add them by adding their numerators.

$$\frac{-64x^{2}+56x+16+16x^{3}+25x\left(x-2\right)}{x-2}=16x^{2}-50$$

Do the multiplications in -64x^{2}+56x+16+16x^{3}+25x\left(x-2\right).

$$\frac{-64x^{2}+56x+16+16x^{3}+25x^{2}-50x}{x-2}=16x^{2}-50$$

Combine like terms in -64x^{2}+56x+16+16x^{3}+25x^{2}-50x.

$$\frac{-39x^{2}+6x+16+16x^{3}}{x-2}=16x^{2}-50$$

Subtract 16x^{2} from both sides.

$$\frac{-39x^{2}+6x+16+16x^{3}}{x-2}-16x^{2}=-50$$

To add or subtract expressions, expand them to make their denominators the same. Multiply -16x^{2} times \frac{x-2}{x-2}.

$$\frac{-39x^{2}+6x+16+16x^{3}}{x-2}+\frac{-16x^{2}\left(x-2\right)}{x-2}=-50$$

Since \frac{-39x^{2}+6x+16+16x^{3}}{x-2} and \frac{-16x^{2}\left(x-2\right)}{x-2} have the same denominator, add them by adding their numerators.

$$\frac{-39x^{2}+6x+16+16x^{3}-16x^{2}\left(x-2\right)}{x-2}=-50$$

Do the multiplications in -39x^{2}+6x+16+16x^{3}-16x^{2}\left(x-2\right).

$$\frac{-39x^{2}+6x+16+16x^{3}-16x^{3}+32x^{2}}{x-2}=-50$$

Combine like terms in -39x^{2}+6x+16+16x^{3}-16x^{3}+32x^{2}.

$$\frac{-7x^{2}+6x+16}{x-2}=-50$$

Add 50 to both sides.

$$\frac{-7x^{2}+6x+16}{x-2}+50=0$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 50 times \frac{x-2}{x-2}.

$$\frac{-7x^{2}+6x+16}{x-2}+\frac{50\left(x-2\right)}{x-2}=0$$

Since \frac{-7x^{2}+6x+16}{x-2} and \frac{50\left(x-2\right)}{x-2} have the same denominator, add them by adding their numerators.

$$\frac{-7x^{2}+6x+16+50\left(x-2\right)}{x-2}=0$$

Do the multiplications in -7x^{2}+6x+16+50\left(x-2\right).

$$\frac{-7x^{2}+6x+16+50x-100}{x-2}=0$$

Combine like terms in -7x^{2}+6x+16+50x-100.

$$\frac{-7x^{2}+56x-84}{x-2}=0$$

Variable x cannot be equal to 2 since division by zero is not defined. Multiply both sides of the equation by x-2.

$$-7x^{2}+56x-84=0$$

Divide both sides by 7.

$$-x^{2}+8x-12=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx-12. To find a and b, set up a system to be solved.

$$a+b=8 ab=-\left(-12\right)=12$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 12.

$$1,12 2,6 3,4$$

Calculate the sum for each pair.

$$1+12=13 2+6=8 3+4=7$$

The solution is the pair that gives sum 8.

$$a=6 b=2$$

Rewrite -x^{2}+8x-12 as \left(-x^{2}+6x\right)+\left(2x-12\right).

$$\left(-x^{2}+6x\right)+\left(2x-12\right)$$

Factor out -x in the first and 2 in the second group.

$$-x\left(x-6\right)+2\left(x-6\right)$$

Factor out common term x-6 by using distributive property.

$$\left(x-6\right)\left(-x+2\right)$$

To find equation solutions, solve x-6=0 and -x+2=0.

$$x=6 x=2$$

Variable x cannot be equal to 2.

$$x=6$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -2,2 since division by zero is not defined. Multiply both sides of the equation by \left(x-2\right)\left(x+2\right), the least common multiple of x+2,x-2.

$$8x\left(x-2\right)\left(x+2\right)+\left(x-2\right)\left(x+2\right)\times 16+\left(x-2\right)\times 8\times \frac{7}{x-2}=\left(x+2\right)\left(8x^{2}-25\right)$$

Use the distributive property to multiply 8x by x-2.

$$\left(8x^{2}-16x\right)\left(x+2\right)+\left(x-2\right)\left(x+2\right)\times 16+\left(x-2\right)\times 8\times \frac{7}{x-2}=\left(x+2\right)\left(8x^{2}-25\right)$$

Use the distributive property to multiply 8x^{2}-16x by x+2 and combine like terms.

$$8x^{3}-32x+\left(x-2\right)\left(x+2\right)\times 16+\left(x-2\right)\times 8\times \frac{7}{x-2}=\left(x+2\right)\left(8x^{2}-25\right)$$

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

$$8x^{3}-32x+\left(x^{2}-4\right)\times 16+\left(x-2\right)\times 8\times \frac{7}{x-2}=\left(x+2\right)\left(8x^{2}-25\right)$$

Use the distributive property to multiply x^{2}-4 by 16.

$$8x^{3}-32x+16x^{2}-64+\left(x-2\right)\times 8\times \frac{7}{x-2}=\left(x+2\right)\left(8x^{2}-25\right)$$

Express \left(x-2\right)\times \frac{7}{x-2} as a single fraction.

$$8x^{3}-32x+16x^{2}-64+\frac{\left(x-2\right)\times 7}{x-2}\times 8=\left(x+2\right)\left(8x^{2}-25\right)$$

Use the distributive property to multiply x+2 by 8x^{2}-25.

$$8x^{3}-32x+16x^{2}-64+\frac{\left(x-2\right)\times 7}{x-2}\times 8=8x^{3}-25x+16x^{2}-50$$

Use the distributive property to multiply x-2 by 7.

$$8x^{3}-32x+16x^{2}-64+\frac{7x-14}{x-2}\times 8=8x^{3}-25x+16x^{2}-50$$

Express \frac{7x-14}{x-2}\times 8 as a single fraction.

$$8x^{3}-32x+16x^{2}-64+\frac{\left(7x-14\right)\times 8}{x-2}=8x^{3}-25x+16x^{2}-50$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 8x^{3}-32x+16x^{2}-64 times \frac{x-2}{x-2}.

$$\frac{\left(8x^{3}-32x+16x^{2}-64\right)\left(x-2\right)}{x-2}+\frac{\left(7x-14\right)\times 8}{x-2}=8x^{3}-25x+16x^{2}-50$$

Since \frac{\left(8x^{3}-32x+16x^{2}-64\right)\left(x-2\right)}{x-2} and \frac{\left(7x-14\right)\times 8}{x-2} have the same denominator, add them by adding their numerators.

$$\frac{\left(8x^{3}-32x+16x^{2}-64\right)\left(x-2\right)+\left(7x-14\right)\times 8}{x-2}=8x^{3}-25x+16x^{2}-50$$

Do the multiplications in \left(8x^{3}-32x+16x^{2}-64\right)\left(x-2\right)+\left(7x-14\right)\times 8.

$$\frac{8x^{4}-16x^{3}-32x^{2}+64x+16x^{3}-32x^{2}-64x+128+56x-112}{x-2}=8x^{3}-25x+16x^{2}-50$$

Combine like terms in 8x^{4}-16x^{3}-32x^{2}+64x+16x^{3}-32x^{2}-64x+128+56x-112.

$$\frac{8x^{4}-64x^{2}+56x+16}{x-2}=8x^{3}-25x+16x^{2}-50$$

Subtract 8x^{3} from both sides.

$$\frac{8x^{4}-64x^{2}+56x+16}{x-2}-8x^{3}=-25x+16x^{2}-50$$

To add or subtract expressions, expand them to make their denominators the same. Multiply -8x^{3} times \frac{x-2}{x-2}.

$$\frac{8x^{4}-64x^{2}+56x+16}{x-2}+\frac{-8x^{3}\left(x-2\right)}{x-2}=-25x+16x^{2}-50$$

Since \frac{8x^{4}-64x^{2}+56x+16}{x-2} and \frac{-8x^{3}\left(x-2\right)}{x-2} have the same denominator, add them by adding their numerators.

$$\frac{8x^{4}-64x^{2}+56x+16-8x^{3}\left(x-2\right)}{x-2}=-25x+16x^{2}-50$$

Do the multiplications in 8x^{4}-64x^{2}+56x+16-8x^{3}\left(x-2\right).

$$\frac{8x^{4}-64x^{2}+56x+16-8x^{4}+16x^{3}}{x-2}=-25x+16x^{2}-50$$

Combine like terms in 8x^{4}-64x^{2}+56x+16-8x^{4}+16x^{3}.

$$\frac{-64x^{2}+56x+16+16x^{3}}{x-2}=-25x+16x^{2}-50$$

Add 25x to both sides.

$$\frac{-64x^{2}+56x+16+16x^{3}}{x-2}+25x=16x^{2}-50$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 25x times \frac{x-2}{x-2}.

$$\frac{-64x^{2}+56x+16+16x^{3}}{x-2}+\frac{25x\left(x-2\right)}{x-2}=16x^{2}-50$$

Since \frac{-64x^{2}+56x+16+16x^{3}}{x-2} and \frac{25x\left(x-2\right)}{x-2} have the same denominator, add them by adding their numerators.

$$\frac{-64x^{2}+56x+16+16x^{3}+25x\left(x-2\right)}{x-2}=16x^{2}-50$$

Do the multiplications in -64x^{2}+56x+16+16x^{3}+25x\left(x-2\right).

$$\frac{-64x^{2}+56x+16+16x^{3}+25x^{2}-50x}{x-2}=16x^{2}-50$$

Combine like terms in -64x^{2}+56x+16+16x^{3}+25x^{2}-50x.

$$\frac{-39x^{2}+6x+16+16x^{3}}{x-2}=16x^{2}-50$$

Subtract 16x^{2} from both sides.

$$\frac{-39x^{2}+6x+16+16x^{3}}{x-2}-16x^{2}=-50$$

To add or subtract expressions, expand them to make their denominators the same. Multiply -16x^{2} times \frac{x-2}{x-2}.

$$\frac{-39x^{2}+6x+16+16x^{3}}{x-2}+\frac{-16x^{2}\left(x-2\right)}{x-2}=-50$$

Since \frac{-39x^{2}+6x+16+16x^{3}}{x-2} and \frac{-16x^{2}\left(x-2\right)}{x-2} have the same denominator, add them by adding their numerators.

$$\frac{-39x^{2}+6x+16+16x^{3}-16x^{2}\left(x-2\right)}{x-2}=-50$$

Do the multiplications in -39x^{2}+6x+16+16x^{3}-16x^{2}\left(x-2\right).

$$\frac{-39x^{2}+6x+16+16x^{3}-16x^{3}+32x^{2}}{x-2}=-50$$

Combine like terms in -39x^{2}+6x+16+16x^{3}-16x^{3}+32x^{2}.

$$\frac{-7x^{2}+6x+16}{x-2}=-50$$

Add 50 to both sides.

$$\frac{-7x^{2}+6x+16}{x-2}+50=0$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 50 times \frac{x-2}{x-2}.

$$\frac{-7x^{2}+6x+16}{x-2}+\frac{50\left(x-2\right)}{x-2}=0$$

Since \frac{-7x^{2}+6x+16}{x-2} and \frac{50\left(x-2\right)}{x-2} have the same denominator, add them by adding their numerators.

$$\frac{-7x^{2}+6x+16+50\left(x-2\right)}{x-2}=0$$

Do the multiplications in -7x^{2}+6x+16+50\left(x-2\right).

$$\frac{-7x^{2}+6x+16+50x-100}{x-2}=0$$

Combine like terms in -7x^{2}+6x+16+50x-100.

$$\frac{-7x^{2}+56x-84}{x-2}=0$$

Variable x cannot be equal to 2 since division by zero is not defined. Multiply both sides of the equation by x-2.

$$-7x^{2}+56x-84=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -7 for a, 56 for b, and -84 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-56±\sqrt{56^{2}-4\left(-7\right)\left(-84\right)}}{2\left(-7\right)}$$

Square 56.

$$x=\frac{-56±\sqrt{3136-4\left(-7\right)\left(-84\right)}}{2\left(-7\right)}$$

Multiply -4 times -7.

$$x=\frac{-56±\sqrt{3136+28\left(-84\right)}}{2\left(-7\right)}$$

Multiply 28 times -84.

$$x=\frac{-56±\sqrt{3136-2352}}{2\left(-7\right)}$$

Add 3136 to -2352.

$$x=\frac{-56±\sqrt{784}}{2\left(-7\right)}$$

Take the square root of 784.

$$x=\frac{-56±28}{2\left(-7\right)}$$

Multiply 2 times -7.

$$x=\frac{-56±28}{-14}$$

Now solve the equation x=\frac{-56±28}{-14} when ± is plus. Add -56 to 28.

$$x=-\frac{28}{-14}$$

Divide -28 by -14.

$$x=2$$

Now solve the equation x=\frac{-56±28}{-14} when ± is minus. Subtract 28 from -56.

$$x=-\frac{84}{-14}$$

Divide -84 by -14.

$$x=6$$

The equation is now solved.

$$x=2 x=6$$

Variable x cannot be equal to 2.

$$x=6$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -2,2 since division by zero is not defined. Multiply both sides of the equation by \left(x-2\right)\left(x+2\right), the least common multiple of x+2,x-2.

$$8x\left(x-2\right)\left(x+2\right)+\left(x-2\right)\left(x+2\right)\times 16+\left(x-2\right)\times 8\times \frac{7}{x-2}=\left(x+2\right)\left(8x^{2}-25\right)$$

Use the distributive property to multiply 8x by x-2.

$$\left(8x^{2}-16x\right)\left(x+2\right)+\left(x-2\right)\left(x+2\right)\times 16+\left(x-2\right)\times 8\times \frac{7}{x-2}=\left(x+2\right)\left(8x^{2}-25\right)$$

Use the distributive property to multiply 8x^{2}-16x by x+2 and combine like terms.

$$8x^{3}-32x+\left(x-2\right)\left(x+2\right)\times 16+\left(x-2\right)\times 8\times \frac{7}{x-2}=\left(x+2\right)\left(8x^{2}-25\right)$$

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

$$8x^{3}-32x+\left(x^{2}-4\right)\times 16+\left(x-2\right)\times 8\times \frac{7}{x-2}=\left(x+2\right)\left(8x^{2}-25\right)$$

Use the distributive property to multiply x^{2}-4 by 16.

$$8x^{3}-32x+16x^{2}-64+\left(x-2\right)\times 8\times \frac{7}{x-2}=\left(x+2\right)\left(8x^{2}-25\right)$$

Express \left(x-2\right)\times \frac{7}{x-2} as a single fraction.

$$8x^{3}-32x+16x^{2}-64+\frac{\left(x-2\right)\times 7}{x-2}\times 8=\left(x+2\right)\left(8x^{2}-25\right)$$

Use the distributive property to multiply x+2 by 8x^{2}-25.

$$8x^{3}-32x+16x^{2}-64+\frac{\left(x-2\right)\times 7}{x-2}\times 8=8x^{3}-25x+16x^{2}-50$$

Use the distributive property to multiply x-2 by 7.

$$8x^{3}-32x+16x^{2}-64+\frac{7x-14}{x-2}\times 8=8x^{3}-25x+16x^{2}-50$$

Express \frac{7x-14}{x-2}\times 8 as a single fraction.

$$8x^{3}-32x+16x^{2}-64+\frac{\left(7x-14\right)\times 8}{x-2}=8x^{3}-25x+16x^{2}-50$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 8x^{3}-32x+16x^{2}-64 times \frac{x-2}{x-2}.

$$\frac{\left(8x^{3}-32x+16x^{2}-64\right)\left(x-2\right)}{x-2}+\frac{\left(7x-14\right)\times 8}{x-2}=8x^{3}-25x+16x^{2}-50$$

Since \frac{\left(8x^{3}-32x+16x^{2}-64\right)\left(x-2\right)}{x-2} and \frac{\left(7x-14\right)\times 8}{x-2} have the same denominator, add them by adding their numerators.

$$\frac{\left(8x^{3}-32x+16x^{2}-64\right)\left(x-2\right)+\left(7x-14\right)\times 8}{x-2}=8x^{3}-25x+16x^{2}-50$$

Do the multiplications in \left(8x^{3}-32x+16x^{2}-64\right)\left(x-2\right)+\left(7x-14\right)\times 8.

$$\frac{8x^{4}-16x^{3}-32x^{2}+64x+16x^{3}-32x^{2}-64x+128+56x-112}{x-2}=8x^{3}-25x+16x^{2}-50$$

Combine like terms in 8x^{4}-16x^{3}-32x^{2}+64x+16x^{3}-32x^{2}-64x+128+56x-112.

$$\frac{8x^{4}-64x^{2}+56x+16}{x-2}=8x^{3}-25x+16x^{2}-50$$

Subtract 8x^{3} from both sides.

$$\frac{8x^{4}-64x^{2}+56x+16}{x-2}-8x^{3}=-25x+16x^{2}-50$$

To add or subtract expressions, expand them to make their denominators the same. Multiply -8x^{3} times \frac{x-2}{x-2}.

$$\frac{8x^{4}-64x^{2}+56x+16}{x-2}+\frac{-8x^{3}\left(x-2\right)}{x-2}=-25x+16x^{2}-50$$

Since \frac{8x^{4}-64x^{2}+56x+16}{x-2} and \frac{-8x^{3}\left(x-2\right)}{x-2} have the same denominator, add them by adding their numerators.

$$\frac{8x^{4}-64x^{2}+56x+16-8x^{3}\left(x-2\right)}{x-2}=-25x+16x^{2}-50$$

Do the multiplications in 8x^{4}-64x^{2}+56x+16-8x^{3}\left(x-2\right).

$$\frac{8x^{4}-64x^{2}+56x+16-8x^{4}+16x^{3}}{x-2}=-25x+16x^{2}-50$$

Combine like terms in 8x^{4}-64x^{2}+56x+16-8x^{4}+16x^{3}.

$$\frac{-64x^{2}+56x+16+16x^{3}}{x-2}=-25x+16x^{2}-50$$

Add 25x to both sides.

$$\frac{-64x^{2}+56x+16+16x^{3}}{x-2}+25x=16x^{2}-50$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 25x times \frac{x-2}{x-2}.

$$\frac{-64x^{2}+56x+16+16x^{3}}{x-2}+\frac{25x\left(x-2\right)}{x-2}=16x^{2}-50$$

Since \frac{-64x^{2}+56x+16+16x^{3}}{x-2} and \frac{25x\left(x-2\right)}{x-2} have the same denominator, add them by adding their numerators.

$$\frac{-64x^{2}+56x+16+16x^{3}+25x\left(x-2\right)}{x-2}=16x^{2}-50$$

Do the multiplications in -64x^{2}+56x+16+16x^{3}+25x\left(x-2\right).

$$\frac{-64x^{2}+56x+16+16x^{3}+25x^{2}-50x}{x-2}=16x^{2}-50$$

Combine like terms in -64x^{2}+56x+16+16x^{3}+25x^{2}-50x.

$$\frac{-39x^{2}+6x+16+16x^{3}}{x-2}=16x^{2}-50$$

Subtract 16x^{2} from both sides.

$$\frac{-39x^{2}+6x+16+16x^{3}}{x-2}-16x^{2}=-50$$

To add or subtract expressions, expand them to make their denominators the same. Multiply -16x^{2} times \frac{x-2}{x-2}.

$$\frac{-39x^{2}+6x+16+16x^{3}}{x-2}+\frac{-16x^{2}\left(x-2\right)}{x-2}=-50$$

Since \frac{-39x^{2}+6x+16+16x^{3}}{x-2} and \frac{-16x^{2}\left(x-2\right)}{x-2} have the same denominator, add them by adding their numerators.

$$\frac{-39x^{2}+6x+16+16x^{3}-16x^{2}\left(x-2\right)}{x-2}=-50$$

Do the multiplications in -39x^{2}+6x+16+16x^{3}-16x^{2}\left(x-2\right).

$$\frac{-39x^{2}+6x+16+16x^{3}-16x^{3}+32x^{2}}{x-2}=-50$$

Combine like terms in -39x^{2}+6x+16+16x^{3}-16x^{3}+32x^{2}.

$$\frac{-7x^{2}+6x+16}{x-2}=-50$$

Variable x cannot be equal to 2 since division by zero is not defined. Multiply both sides of the equation by x-2.

$$-7x^{2}+6x+16=-50\left(x-2\right)$$

Use the distributive property to multiply -50 by x-2.

$$-7x^{2}+6x+16=-50x+100$$

Add 50x to both sides.

$$-7x^{2}+6x+16+50x=100$$

Combine 6x and 50x to get 56x.

$$-7x^{2}+56x+16=100$$

Subtract 16 from both sides.

$$-7x^{2}+56x=100-16$$

Subtract 16 from 100 to get 84.

$$-7x^{2}+56x=84$$

Divide both sides by -7.

$$\frac{-7x^{2}+56x}{-7}=\frac{84}{-7}$$

Dividing by -7 undoes the multiplication by -7.

$$x^{2}+\frac{56}{-7}x=\frac{84}{-7}$$

Divide 56 by -7.

$$x^{2}-8x=\frac{84}{-7}$$

Divide 84 by -7.

$$x^{2}-8x=-12$$

Divide -8, the coefficient of the x term, by 2 to get -4. Then add the square of -4 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-8x+\left(-4\right)^{2}=-12+\left(-4\right)^{2}$$

Square -4.

$$x^{2}-8x+16=-12+16$$

Add -12 to 16.

$$x^{2}-8x+16=4$$

Factor x^{2}-8x+16. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-4\right)^{2}=4$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-4\right)^{2}}=\sqrt{4}$$

Simplify.

$$x-4=2 x-4=-2$$

Add 4 to both sides of the equation.

$$x=6 x=2$$

Variable x cannot be equal to 2.

$$x=6$$