Solve frac{18x^{2}+3}{32x^{3}-6x-1}-x^2+1/8x^3-2x^2-x+1/x+9x+5/16x^2+8x+1

$$\frac { 18 x ^ { 2 } + 3 } { 32 x ^ { 3 } - 6 x - 1 } - \frac { x ^ { 2 } + 1 } { 8 x ^ { 3 } - 2 x ^ { 2 } - x } + \frac { 1 } { x } + \frac { 9 x + 5 } { 16 x ^ { 2 } + 8 x + 1 }$$

Evaluate

$$\frac{2}{x}$$

Short Solution Steps

Factor 32x^{3}-6x-1. Factor 8x^{3}-2x^{2}-x.

$$\frac{18x^{2}+3}{\left(2x-1\right)\left(4x+1\right)^{2}}-\frac{x^{2}+1}{x\left(2x-1\right)\left(4x+1\right)}+\frac{1}{x}+\frac{9x+5}{16x^{2}+8x+1}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(2x-1\right)\left(4x+1\right)^{2} and x\left(2x-1\right)\left(4x+1\right) is x\left(2x-1\right)\left(4x+1\right)^{2}. Multiply \frac{18x^{2}+3}{\left(2x-1\right)\left(4x+1\right)^{2}} times \frac{x}{x}. Multiply \frac{x^{2}+1}{x\left(2x-1\right)\left(4x+1\right)} times \frac{4x+1}{4x+1}.

$$\frac{\left(18x^{2}+3\right)x}{x\left(2x-1\right)\left(4x+1\right)^{2}}-\frac{\left(x^{2}+1\right)\left(4x+1\right)}{x\left(2x-1\right)\left(4x+1\right)^{2}}+\frac{1}{x}+\frac{9x+5}{16x^{2}+8x+1}$$

Since \frac{\left(18x^{2}+3\right)x}{x\left(2x-1\right)\left(4x+1\right)^{2}} and \frac{\left(x^{2}+1\right)\left(4x+1\right)}{x\left(2x-1\right)\left(4x+1\right)^{2}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(18x^{2}+3\right)x-\left(x^{2}+1\right)\left(4x+1\right)}{x\left(2x-1\right)\left(4x+1\right)^{2}}+\frac{1}{x}+\frac{9x+5}{16x^{2}+8x+1}$$

Do the multiplications in \left(18x^{2}+3\right)x-\left(x^{2}+1\right)\left(4x+1\right).

$$\frac{18x^{3}+3x-4x^{3}-x^{2}-4x-1}{x\left(2x-1\right)\left(4x+1\right)^{2}}+\frac{1}{x}+\frac{9x+5}{16x^{2}+8x+1}$$

Combine like terms in 18x^{3}+3x-4x^{3}-x^{2}-4x-1.

$$\frac{14x^{3}-x-x^{2}-1}{x\left(2x-1\right)\left(4x+1\right)^{2}}+\frac{1}{x}+\frac{9x+5}{16x^{2}+8x+1}$$

Factor the expressions that are not already factored in \frac{14x^{3}-x-x^{2}-1}{x\left(2x-1\right)\left(4x+1\right)^{2}}.

$$\frac{\left(2x-1\right)\left(7x^{2}+3x+1\right)}{x\left(2x-1\right)\left(4x+1\right)^{2}}+\frac{1}{x}+\frac{9x+5}{16x^{2}+8x+1}$$

Cancel out 2x-1 in both numerator and denominator.

$$\frac{7x^{2}+3x+1}{x\left(4x+1\right)^{2}}+\frac{1}{x}+\frac{9x+5}{16x^{2}+8x+1}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x\left(4x+1\right)^{2} and x is x\left(4x+1\right)^{2}. Multiply \frac{1}{x} times \frac{\left(4x+1\right)^{2}}{\left(4x+1\right)^{2}}.

$$\frac{7x^{2}+3x+1}{x\left(4x+1\right)^{2}}+\frac{\left(4x+1\right)^{2}}{x\left(4x+1\right)^{2}}+\frac{9x+5}{16x^{2}+8x+1}$$

Since \frac{7x^{2}+3x+1}{x\left(4x+1\right)^{2}} and \frac{\left(4x+1\right)^{2}}{x\left(4x+1\right)^{2}} have the same denominator, add them by adding their numerators.

$$\frac{7x^{2}+3x+1+\left(4x+1\right)^{2}}{x\left(4x+1\right)^{2}}+\frac{9x+5}{16x^{2}+8x+1}$$

Do the multiplications in 7x^{2}+3x+1+\left(4x+1\right)^{2}.

$$\frac{7x^{2}+3x+1+16x^{2}+8x+1}{x\left(4x+1\right)^{2}}+\frac{9x+5}{16x^{2}+8x+1}$$

Combine like terms in 7x^{2}+3x+1+16x^{2}+8x+1.

$$\frac{23x^{2}+11x+2}{x\left(4x+1\right)^{2}}+\frac{9x+5}{16x^{2}+8x+1}$$

Factor 16x^{2}+8x+1.

$$\frac{23x^{2}+11x+2}{x\left(4x+1\right)^{2}}+\frac{9x+5}{\left(4x+1\right)^{2}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x\left(4x+1\right)^{2} and \left(4x+1\right)^{2} is x\left(4x+1\right)^{2}. Multiply \frac{9x+5}{\left(4x+1\right)^{2}} times \frac{x}{x}.

$$\frac{23x^{2}+11x+2}{x\left(4x+1\right)^{2}}+\frac{\left(9x+5\right)x}{x\left(4x+1\right)^{2}}$$

Since \frac{23x^{2}+11x+2}{x\left(4x+1\right)^{2}} and \frac{\left(9x+5\right)x}{x\left(4x+1\right)^{2}} have the same denominator, add them by adding their numerators.

$$\frac{23x^{2}+11x+2+\left(9x+5\right)x}{x\left(4x+1\right)^{2}}$$

Do the multiplications in 23x^{2}+11x+2+\left(9x+5\right)x.

$$\frac{23x^{2}+11x+2+9x^{2}+5x}{x\left(4x+1\right)^{2}}$$

Combine like terms in 23x^{2}+11x+2+9x^{2}+5x.

$$\frac{32x^{2}+16x+2}{x\left(4x+1\right)^{2}}$$

Factor the expressions that are not already factored in \frac{32x^{2}+16x+2}{x\left(4x+1\right)^{2}}.

$$\frac{2\left(4x+1\right)^{2}}{x\left(4x+1\right)^{2}}$$

Cancel out \left(4x+1\right)^{2} in both numerator and denominator.

$$\frac{2}{x}$$

Expand

$$\frac{2}{x}$$

Short Solution Steps

Factor 32x^{3}-6x-1. Factor 8x^{3}-2x^{2}-x.

$$\frac{18x^{2}+3}{\left(2x-1\right)\left(4x+1\right)^{2}}-\frac{x^{2}+1}{x\left(2x-1\right)\left(4x+1\right)}+\frac{1}{x}+\frac{9x+5}{16x^{2}+8x+1}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(2x-1\right)\left(4x+1\right)^{2} and x\left(2x-1\right)\left(4x+1\right) is x\left(2x-1\right)\left(4x+1\right)^{2}. Multiply \frac{18x^{2}+3}{\left(2x-1\right)\left(4x+1\right)^{2}} times \frac{x}{x}. Multiply \frac{x^{2}+1}{x\left(2x-1\right)\left(4x+1\right)} times \frac{4x+1}{4x+1}.

$$\frac{\left(18x^{2}+3\right)x}{x\left(2x-1\right)\left(4x+1\right)^{2}}-\frac{\left(x^{2}+1\right)\left(4x+1\right)}{x\left(2x-1\right)\left(4x+1\right)^{2}}+\frac{1}{x}+\frac{9x+5}{16x^{2}+8x+1}$$

Since \frac{\left(18x^{2}+3\right)x}{x\left(2x-1\right)\left(4x+1\right)^{2}} and \frac{\left(x^{2}+1\right)\left(4x+1\right)}{x\left(2x-1\right)\left(4x+1\right)^{2}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(18x^{2}+3\right)x-\left(x^{2}+1\right)\left(4x+1\right)}{x\left(2x-1\right)\left(4x+1\right)^{2}}+\frac{1}{x}+\frac{9x+5}{16x^{2}+8x+1}$$

Do the multiplications in \left(18x^{2}+3\right)x-\left(x^{2}+1\right)\left(4x+1\right).

$$\frac{18x^{3}+3x-4x^{3}-x^{2}-4x-1}{x\left(2x-1\right)\left(4x+1\right)^{2}}+\frac{1}{x}+\frac{9x+5}{16x^{2}+8x+1}$$

Combine like terms in 18x^{3}+3x-4x^{3}-x^{2}-4x-1.

$$\frac{14x^{3}-x-x^{2}-1}{x\left(2x-1\right)\left(4x+1\right)^{2}}+\frac{1}{x}+\frac{9x+5}{16x^{2}+8x+1}$$

Factor the expressions that are not already factored in \frac{14x^{3}-x-x^{2}-1}{x\left(2x-1\right)\left(4x+1\right)^{2}}.

$$\frac{\left(2x-1\right)\left(7x^{2}+3x+1\right)}{x\left(2x-1\right)\left(4x+1\right)^{2}}+\frac{1}{x}+\frac{9x+5}{16x^{2}+8x+1}$$

Cancel out 2x-1 in both numerator and denominator.

$$\frac{7x^{2}+3x+1}{x\left(4x+1\right)^{2}}+\frac{1}{x}+\frac{9x+5}{16x^{2}+8x+1}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x\left(4x+1\right)^{2} and x is x\left(4x+1\right)^{2}. Multiply \frac{1}{x} times \frac{\left(4x+1\right)^{2}}{\left(4x+1\right)^{2}}.

$$\frac{7x^{2}+3x+1}{x\left(4x+1\right)^{2}}+\frac{\left(4x+1\right)^{2}}{x\left(4x+1\right)^{2}}+\frac{9x+5}{16x^{2}+8x+1}$$

Since \frac{7x^{2}+3x+1}{x\left(4x+1\right)^{2}} and \frac{\left(4x+1\right)^{2}}{x\left(4x+1\right)^{2}} have the same denominator, add them by adding their numerators.

$$\frac{7x^{2}+3x+1+\left(4x+1\right)^{2}}{x\left(4x+1\right)^{2}}+\frac{9x+5}{16x^{2}+8x+1}$$

Do the multiplications in 7x^{2}+3x+1+\left(4x+1\right)^{2}.

$$\frac{7x^{2}+3x+1+16x^{2}+8x+1}{x\left(4x+1\right)^{2}}+\frac{9x+5}{16x^{2}+8x+1}$$

Combine like terms in 7x^{2}+3x+1+16x^{2}+8x+1.

$$\frac{23x^{2}+11x+2}{x\left(4x+1\right)^{2}}+\frac{9x+5}{16x^{2}+8x+1}$$

Factor 16x^{2}+8x+1.

$$\frac{23x^{2}+11x+2}{x\left(4x+1\right)^{2}}+\frac{9x+5}{\left(4x+1\right)^{2}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x\left(4x+1\right)^{2} and \left(4x+1\right)^{2} is x\left(4x+1\right)^{2}. Multiply \frac{9x+5}{\left(4x+1\right)^{2}} times \frac{x}{x}.

$$\frac{23x^{2}+11x+2}{x\left(4x+1\right)^{2}}+\frac{\left(9x+5\right)x}{x\left(4x+1\right)^{2}}$$

Since \frac{23x^{2}+11x+2}{x\left(4x+1\right)^{2}} and \frac{\left(9x+5\right)x}{x\left(4x+1\right)^{2}} have the same denominator, add them by adding their numerators.

$$\frac{23x^{2}+11x+2+\left(9x+5\right)x}{x\left(4x+1\right)^{2}}$$

Do the multiplications in 23x^{2}+11x+2+\left(9x+5\right)x.

$$\frac{23x^{2}+11x+2+9x^{2}+5x}{x\left(4x+1\right)^{2}}$$

Combine like terms in 23x^{2}+11x+2+9x^{2}+5x.

$$\frac{32x^{2}+16x+2}{x\left(4x+1\right)^{2}}$$

Factor the expressions that are not already factored in \frac{32x^{2}+16x+2}{x\left(4x+1\right)^{2}}.

$$\frac{2\left(4x+1\right)^{2}}{x\left(4x+1\right)^{2}}$$

Cancel out \left(4x+1\right)^{2} in both numerator and denominator.

$$\frac{2}{x}$$