Solve 2x/x^2+4x+4+x-1/x(x+2)

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

Evaluate

$$\frac{\left(3x-2\right)\left(x+1\right)}{x\left(x+2\right)^{2}}$$

Short Solution Steps

Factor x^{2}+4x+4.

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

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

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

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

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

Do the multiplications in 2xx+\left(x-1\right)\left(x+2\right).

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

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

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

Expand x\left(x+2\right)^{2}.

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

Expand

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

Short Solution Steps

Factor x^{2}+4x+4.

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

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

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

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

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

Do the multiplications in 2xx+\left(x-1\right)\left(x+2\right).

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

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

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

Expand x\left(x+2\right)^{2}.

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