Solve 8x+16+8/x+2*1/x-2=8x^2-25/x-2

$$8 x + 16 + \frac { 8 } { x + 2 } \cdot \frac { 1 } { x - 2 } = \frac { 8 x ^ { 2 } - 25 } { x - 2 }$$

Solve for x

$$x=-\frac{6}{7}\approx -0.857142857$$

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.

$$8x\left(x-2\right)\left(x+2\right)+\left(x-2\right)\left(x+2\right)\times 16+\left(x-2\right)\times 8\times \frac{1}{x-2}=\left(x+2\right)\left(8x^{2}-25\right)$$

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

$$\left(8x^{2}-16x\right)\left(x+2\right)+\left(x-2\right)\left(x+2\right)\times 16+\left(x-2\right)\times 8\times \frac{1}{x-2}=\left(x+2\right)\left(8x^{2}-25\right)$$

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

$$8x^{3}-32x+\left(x-2\right)\left(x+2\right)\times 16+\left(x-2\right)\times 8\times \frac{1}{x-2}=\left(x+2\right)\left(8x^{2}-25\right)$$

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

$$8x^{3}-32x+\left(x^{2}-4\right)\times 16+\left(x-2\right)\times 8\times \frac{1}{x-2}=\left(x+2\right)\left(8x^{2}-25\right)$$

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

$$8x^{3}-32x+16x^{2}-64+\left(x-2\right)\times 8\times \frac{1}{x-2}=\left(x+2\right)\left(8x^{2}-25\right)$$

Express \left(x-2\right)\times \frac{1}{x-2} as a single fraction.

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

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

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

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

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

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}.

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

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

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

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

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

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

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

Subtract 8x^{3} from both sides.

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

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

$$\frac{8x^{4}-64x^{2}+8x+112}{x-2}+\frac{-8x^{3}\left(x-2\right)}{x-2}=-25x+16x^{2}-50$$

Since \frac{8x^{4}-64x^{2}+8x+112}{x-2} and \frac{-8x^{3}\left(x-2\right)}{x-2} have the same denominator, add them by adding their numerators.

$$\frac{8x^{4}-64x^{2}+8x+112-8x^{3}\left(x-2\right)}{x-2}=-25x+16x^{2}-50$$

Do the multiplications in 8x^{4}-64x^{2}+8x+112-8x^{3}\left(x-2\right).

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

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

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

Add 25x to both sides.

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

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

$$\frac{-64x^{2}+8x+112+16x^{3}}{x-2}+\frac{25x\left(x-2\right)}{x-2}=16x^{2}-50$$

Since \frac{-64x^{2}+8x+112+16x^{3}}{x-2} and \frac{25x\left(x-2\right)}{x-2} have the same denominator, add them by adding their numerators.

$$\frac{-64x^{2}+8x+112+16x^{3}+25x\left(x-2\right)}{x-2}=16x^{2}-50$$

Do the multiplications in -64x^{2}+8x+112+16x^{3}+25x\left(x-2\right).

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

Combine like terms in -64x^{2}+8x+112+16x^{3}+25x^{2}-50x.

$$\frac{-39x^{2}-42x+112+16x^{3}}{x-2}=16x^{2}-50$$

Subtract 16x^{2} from both sides.

$$\frac{-39x^{2}-42x+112+16x^{3}}{x-2}-16x^{2}=-50$$

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

$$\frac{-39x^{2}-42x+112+16x^{3}}{x-2}+\frac{-16x^{2}\left(x-2\right)}{x-2}=-50$$

Since \frac{-39x^{2}-42x+112+16x^{3}}{x-2} and \frac{-16x^{2}\left(x-2\right)}{x-2} have the same denominator, add them by adding their numerators.

$$\frac{-39x^{2}-42x+112+16x^{3}-16x^{2}\left(x-2\right)}{x-2}=-50$$

Do the multiplications in -39x^{2}-42x+112+16x^{3}-16x^{2}\left(x-2\right).

$$\frac{-39x^{2}-42x+112+16x^{3}-16x^{3}+32x^{2}}{x-2}=-50$$

Combine like terms in -39x^{2}-42x+112+16x^{3}-16x^{3}+32x^{2}.

$$\frac{-7x^{2}-42x+112}{x-2}=-50$$

Add 50 to both sides.

$$\frac{-7x^{2}-42x+112}{x-2}+50=0$$

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

$$\frac{-7x^{2}-42x+112}{x-2}+\frac{50\left(x-2\right)}{x-2}=0$$

Since \frac{-7x^{2}-42x+112}{x-2} and \frac{50\left(x-2\right)}{x-2} have the same denominator, add them by adding their numerators.

$$\frac{-7x^{2}-42x+112+50\left(x-2\right)}{x-2}=0$$

Do the multiplications in -7x^{2}-42x+112+50\left(x-2\right).

$$\frac{-7x^{2}-42x+112+50x-100}{x-2}=0$$

Combine like terms in -7x^{2}-42x+112+50x-100.

$$\frac{-7x^{2}+8x+12}{x-2}=0$$

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

$$-7x^{2}+8x+12=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -7x^{2}+ax+bx+12. To find a and b, set up a system to be solved.

$$a+b=8 ab=-7\times 12=-84$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -84.

$$-1,84 -2,42 -3,28 -4,21 -6,14 -7,12$$

Calculate the sum for each pair.

$$-1+84=83 -2+42=40 -3+28=25 -4+21=17 -6+14=8 -7+12=5$$

The solution is the pair that gives sum 8.

$$a=14 b=-6$$

Rewrite -7x^{2}+8x+12 as \left(-7x^{2}+14x\right)+\left(-6x+12\right).

$$\left(-7x^{2}+14x\right)+\left(-6x+12\right)$$

Factor out 7x in the first and 6 in the second group.

$$7x\left(-x+2\right)+6\left(-x+2\right)$$

Factor out common term -x+2 by using distributive property.

$$\left(-x+2\right)\left(7x+6\right)$$

To find equation solutions, solve -x+2=0 and 7x+6=0.

$$x=2 x=-\frac{6}{7}$$

Variable x cannot be equal to 2.

$$x=-\frac{6}{7}$$

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.

$$8x\left(x-2\right)\left(x+2\right)+\left(x-2\right)\left(x+2\right)\times 16+\left(x-2\right)\times 8\times \frac{1}{x-2}=\left(x+2\right)\left(8x^{2}-25\right)$$

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

$$\left(8x^{2}-16x\right)\left(x+2\right)+\left(x-2\right)\left(x+2\right)\times 16+\left(x-2\right)\times 8\times \frac{1}{x-2}=\left(x+2\right)\left(8x^{2}-25\right)$$

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

$$8x^{3}-32x+\left(x-2\right)\left(x+2\right)\times 16+\left(x-2\right)\times 8\times \frac{1}{x-2}=\left(x+2\right)\left(8x^{2}-25\right)$$

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

$$8x^{3}-32x+\left(x^{2}-4\right)\times 16+\left(x-2\right)\times 8\times \frac{1}{x-2}=\left(x+2\right)\left(8x^{2}-25\right)$$

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

$$8x^{3}-32x+16x^{2}-64+\left(x-2\right)\times 8\times \frac{1}{x-2}=\left(x+2\right)\left(8x^{2}-25\right)$$

Express \left(x-2\right)\times \frac{1}{x-2} as a single fraction.

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

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

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

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

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

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}.

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

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

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

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

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

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

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

Subtract 8x^{3} from both sides.

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

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

$$\frac{8x^{4}-64x^{2}+8x+112}{x-2}+\frac{-8x^{3}\left(x-2\right)}{x-2}=-25x+16x^{2}-50$$

Since \frac{8x^{4}-64x^{2}+8x+112}{x-2} and \frac{-8x^{3}\left(x-2\right)}{x-2} have the same denominator, add them by adding their numerators.

$$\frac{8x^{4}-64x^{2}+8x+112-8x^{3}\left(x-2\right)}{x-2}=-25x+16x^{2}-50$$

Do the multiplications in 8x^{4}-64x^{2}+8x+112-8x^{3}\left(x-2\right).

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

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

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

Add 25x to both sides.

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

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

$$\frac{-64x^{2}+8x+112+16x^{3}}{x-2}+\frac{25x\left(x-2\right)}{x-2}=16x^{2}-50$$

Since \frac{-64x^{2}+8x+112+16x^{3}}{x-2} and \frac{25x\left(x-2\right)}{x-2} have the same denominator, add them by adding their numerators.

$$\frac{-64x^{2}+8x+112+16x^{3}+25x\left(x-2\right)}{x-2}=16x^{2}-50$$

Do the multiplications in -64x^{2}+8x+112+16x^{3}+25x\left(x-2\right).

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

Combine like terms in -64x^{2}+8x+112+16x^{3}+25x^{2}-50x.

$$\frac{-39x^{2}-42x+112+16x^{3}}{x-2}=16x^{2}-50$$

Subtract 16x^{2} from both sides.

$$\frac{-39x^{2}-42x+112+16x^{3}}{x-2}-16x^{2}=-50$$

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

$$\frac{-39x^{2}-42x+112+16x^{3}}{x-2}+\frac{-16x^{2}\left(x-2\right)}{x-2}=-50$$

Since \frac{-39x^{2}-42x+112+16x^{3}}{x-2} and \frac{-16x^{2}\left(x-2\right)}{x-2} have the same denominator, add them by adding their numerators.

$$\frac{-39x^{2}-42x+112+16x^{3}-16x^{2}\left(x-2\right)}{x-2}=-50$$

Do the multiplications in -39x^{2}-42x+112+16x^{3}-16x^{2}\left(x-2\right).

$$\frac{-39x^{2}-42x+112+16x^{3}-16x^{3}+32x^{2}}{x-2}=-50$$

Combine like terms in -39x^{2}-42x+112+16x^{3}-16x^{3}+32x^{2}.

$$\frac{-7x^{2}-42x+112}{x-2}=-50$$

Add 50 to both sides.

$$\frac{-7x^{2}-42x+112}{x-2}+50=0$$

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

$$\frac{-7x^{2}-42x+112}{x-2}+\frac{50\left(x-2\right)}{x-2}=0$$

Since \frac{-7x^{2}-42x+112}{x-2} and \frac{50\left(x-2\right)}{x-2} have the same denominator, add them by adding their numerators.

$$\frac{-7x^{2}-42x+112+50\left(x-2\right)}{x-2}=0$$

Do the multiplications in -7x^{2}-42x+112+50\left(x-2\right).

$$\frac{-7x^{2}-42x+112+50x-100}{x-2}=0$$

Combine like terms in -7x^{2}-42x+112+50x-100.

$$\frac{-7x^{2}+8x+12}{x-2}=0$$

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

$$-7x^{2}+8x+12=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -7 for a, 8 for b, and 12 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-8±\sqrt{8^{2}-4\left(-7\right)\times 12}}{2\left(-7\right)}$$

Square 8.

$$x=\frac{-8±\sqrt{64-4\left(-7\right)\times 12}}{2\left(-7\right)}$$

Multiply -4 times -7.

$$x=\frac{-8±\sqrt{64+28\times 12}}{2\left(-7\right)}$$

Multiply 28 times 12.

$$x=\frac{-8±\sqrt{64+336}}{2\left(-7\right)}$$

Add 64 to 336.

$$x=\frac{-8±\sqrt{400}}{2\left(-7\right)}$$

Take the square root of 400.

$$x=\frac{-8±20}{2\left(-7\right)}$$

Multiply 2 times -7.

$$x=\frac{-8±20}{-14}$$

Now solve the equation x=\frac{-8±20}{-14} when ± is plus. Add -8 to 20.

$$x=\frac{12}{-14}$$

Reduce the fraction \frac{12}{-14} to lowest terms by extracting and canceling out 2.

$$x=-\frac{6}{7}$$

Now solve the equation x=\frac{-8±20}{-14} when ± is minus. Subtract 20 from -8.

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

Divide -28 by -14.

$$x=2$$

The equation is now solved.

$$x=-\frac{6}{7} x=2$$

Variable x cannot be equal to 2.

$$x=-\frac{6}{7}$$

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.

$$8x\left(x-2\right)\left(x+2\right)+\left(x-2\right)\left(x+2\right)\times 16+\left(x-2\right)\times 8\times \frac{1}{x-2}=\left(x+2\right)\left(8x^{2}-25\right)$$

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

$$\left(8x^{2}-16x\right)\left(x+2\right)+\left(x-2\right)\left(x+2\right)\times 16+\left(x-2\right)\times 8\times \frac{1}{x-2}=\left(x+2\right)\left(8x^{2}-25\right)$$

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

$$8x^{3}-32x+\left(x-2\right)\left(x+2\right)\times 16+\left(x-2\right)\times 8\times \frac{1}{x-2}=\left(x+2\right)\left(8x^{2}-25\right)$$

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

$$8x^{3}-32x+\left(x^{2}-4\right)\times 16+\left(x-2\right)\times 8\times \frac{1}{x-2}=\left(x+2\right)\left(8x^{2}-25\right)$$

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

$$8x^{3}-32x+16x^{2}-64+\left(x-2\right)\times 8\times \frac{1}{x-2}=\left(x+2\right)\left(8x^{2}-25\right)$$

Express \left(x-2\right)\times \frac{1}{x-2} as a single fraction.

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

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

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

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

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

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}.

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

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

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

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

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

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

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

Subtract 8x^{3} from both sides.

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

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

$$\frac{8x^{4}-64x^{2}+8x+112}{x-2}+\frac{-8x^{3}\left(x-2\right)}{x-2}=-25x+16x^{2}-50$$

Since \frac{8x^{4}-64x^{2}+8x+112}{x-2} and \frac{-8x^{3}\left(x-2\right)}{x-2} have the same denominator, add them by adding their numerators.

$$\frac{8x^{4}-64x^{2}+8x+112-8x^{3}\left(x-2\right)}{x-2}=-25x+16x^{2}-50$$

Do the multiplications in 8x^{4}-64x^{2}+8x+112-8x^{3}\left(x-2\right).

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

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

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

Add 25x to both sides.

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

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

$$\frac{-64x^{2}+8x+112+16x^{3}}{x-2}+\frac{25x\left(x-2\right)}{x-2}=16x^{2}-50$$

Since \frac{-64x^{2}+8x+112+16x^{3}}{x-2} and \frac{25x\left(x-2\right)}{x-2} have the same denominator, add them by adding their numerators.

$$\frac{-64x^{2}+8x+112+16x^{3}+25x\left(x-2\right)}{x-2}=16x^{2}-50$$

Do the multiplications in -64x^{2}+8x+112+16x^{3}+25x\left(x-2\right).

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

Combine like terms in -64x^{2}+8x+112+16x^{3}+25x^{2}-50x.

$$\frac{-39x^{2}-42x+112+16x^{3}}{x-2}=16x^{2}-50$$

Subtract 16x^{2} from both sides.

$$\frac{-39x^{2}-42x+112+16x^{3}}{x-2}-16x^{2}=-50$$

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

$$\frac{-39x^{2}-42x+112+16x^{3}}{x-2}+\frac{-16x^{2}\left(x-2\right)}{x-2}=-50$$

Since \frac{-39x^{2}-42x+112+16x^{3}}{x-2} and \frac{-16x^{2}\left(x-2\right)}{x-2} have the same denominator, add them by adding their numerators.

$$\frac{-39x^{2}-42x+112+16x^{3}-16x^{2}\left(x-2\right)}{x-2}=-50$$

Do the multiplications in -39x^{2}-42x+112+16x^{3}-16x^{2}\left(x-2\right).

$$\frac{-39x^{2}-42x+112+16x^{3}-16x^{3}+32x^{2}}{x-2}=-50$$

Combine like terms in -39x^{2}-42x+112+16x^{3}-16x^{3}+32x^{2}.

$$\frac{-7x^{2}-42x+112}{x-2}=-50$$

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

$$-7x^{2}-42x+112=-50\left(x-2\right)$$

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

$$-7x^{2}-42x+112=-50x+100$$

Add 50x to both sides.

$$-7x^{2}-42x+112+50x=100$$

Combine -42x and 50x to get 8x.

$$-7x^{2}+8x+112=100$$

Subtract 112 from both sides.

$$-7x^{2}+8x=100-112$$

Subtract 112 from 100 to get -12.

$$-7x^{2}+8x=-12$$

Divide both sides by -7.

$$\frac{-7x^{2}+8x}{-7}=-\frac{12}{-7}$$

Dividing by -7 undoes the multiplication by -7.

$$x^{2}+\frac{8}{-7}x=-\frac{12}{-7}$$

Divide 8 by -7.

$$x^{2}-\frac{8}{7}x=-\frac{12}{-7}$$

Divide -12 by -7.

$$x^{2}-\frac{8}{7}x=\frac{12}{7}$$

Divide -\frac{8}{7}, the coefficient of the x term, by 2 to get -\frac{4}{7}. Then add the square of -\frac{4}{7} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{8}{7}x+\left(-\frac{4}{7}\right)^{2}=\frac{12}{7}+\left(-\frac{4}{7}\right)^{2}$$

Square -\frac{4}{7} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{8}{7}x+\frac{16}{49}=\frac{12}{7}+\frac{16}{49}$$

Add \frac{12}{7} to \frac{16}{49} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{8}{7}x+\frac{16}{49}=\frac{100}{49}$$

Factor x^{2}-\frac{8}{7}x+\frac{16}{49}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{4}{7}\right)^{2}=\frac{100}{49}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{4}{7}\right)^{2}}=\sqrt{\frac{100}{49}}$$

Simplify.

$$x-\frac{4}{7}=\frac{10}{7} x-\frac{4}{7}=-\frac{10}{7}$$

Add \frac{4}{7} to both sides of the equation.

$$x=2 x=-\frac{6}{7}$$

Variable x cannot be equal to 2.

$$x=-\frac{6}{7}$$