Solve x/3+frac{5{x}}-4/1+frac{2{x}}

$$\frac { x } { 3 + \frac { 5 } { x } } - \frac { 4 } { 1 + \frac { 2 } { x } }$$

Evaluate

$$\frac{x\left(x^{2}-10x-20\right)}{\left(x+2\right)\left(3x+5\right)}$$

Short Solution Steps

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

$$\frac{x}{\frac{3x}{x}+\frac{5}{x}}-\frac{4}{1+\frac{2}{x}}$$

Since \frac{3x}{x} and \frac{5}{x} have the same denominator, add them by adding their numerators.

$$\frac{x}{\frac{3x+5}{x}}-\frac{4}{1+\frac{2}{x}}$$

Divide x by \frac{3x+5}{x} by multiplying x by the reciprocal of \frac{3x+5}{x}.

$$\frac{xx}{3x+5}-\frac{4}{1+\frac{2}{x}}$$

Multiply x and x to get x^{2}.

$$\frac{x^{2}}{3x+5}-\frac{4}{1+\frac{2}{x}}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{x}{x}.

$$\frac{x^{2}}{3x+5}-\frac{4}{\frac{x}{x}+\frac{2}{x}}$$

Since \frac{x}{x} and \frac{2}{x} have the same denominator, add them by adding their numerators.

$$\frac{x^{2}}{3x+5}-\frac{4}{\frac{x+2}{x}}$$

Divide 4 by \frac{x+2}{x} by multiplying 4 by the reciprocal of \frac{x+2}{x}.

$$\frac{x^{2}}{3x+5}-\frac{4x}{x+2}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 3x+5 and x+2 is \left(x+2\right)\left(3x+5\right). Multiply \frac{x^{2}}{3x+5} times \frac{x+2}{x+2}. Multiply \frac{4x}{x+2} times \frac{3x+5}{3x+5}.

$$\frac{x^{2}\left(x+2\right)}{\left(x+2\right)\left(3x+5\right)}-\frac{4x\left(3x+5\right)}{\left(x+2\right)\left(3x+5\right)}$$

Since \frac{x^{2}\left(x+2\right)}{\left(x+2\right)\left(3x+5\right)} and \frac{4x\left(3x+5\right)}{\left(x+2\right)\left(3x+5\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{x^{2}\left(x+2\right)-4x\left(3x+5\right)}{\left(x+2\right)\left(3x+5\right)}$$

Do the multiplications in x^{2}\left(x+2\right)-4x\left(3x+5\right).

$$\frac{x^{3}+2x^{2}-12x^{2}-20x}{\left(x+2\right)\left(3x+5\right)}$$

Combine like terms in x^{3}+2x^{2}-12x^{2}-20x.

$$\frac{x^{3}-10x^{2}-20x}{\left(x+2\right)\left(3x+5\right)}$$

Expand \left(x+2\right)\left(3x+5\right).

$$\frac{x^{3}-10x^{2}-20x}{3x^{2}+11x+10}$$

Differentiate w.r.t. x

$$\frac{3x^{4}+22x^{3}-20x^{2}-200x-200}{\left(\left(x+2\right)\left(3x+5\right)\right)^{2}}$$