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

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

Evaluate

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

Short Solution Steps

Factor the expressions that are not already factored in \frac{a^{2}-4}{a^{2}-4a+4}.

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

Cancel out a-2 in both numerator and denominator.

$$\frac{a+2}{a-2}-\frac{1}{2-a}+\frac{2}{a^{2}-2a}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of a-2 and 2-a is a-2. Multiply \frac{1}{2-a} times \frac{-1}{-1}.

$$\frac{a+2}{a-2}-\frac{-1}{a-2}+\frac{2}{a^{2}-2a}$$

Since \frac{a+2}{a-2} and \frac{-1}{a-2} have the same denominator, subtract them by subtracting their numerators.

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

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

$$\frac{a+2+1}{a-2}+\frac{2}{a^{2}-2a}$$

Combine like terms in a+2+1.

$$\frac{a+3}{a-2}+\frac{2}{a^{2}-2a}$$

Factor a^{2}-2a.

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

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

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

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

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

Do the multiplications in \left(a+3\right)a+2.

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

Expand a\left(a-2\right).

$$\frac{a^{2}+3a+2}{a^{2}-2a}$$

Expand

$$\frac{a^{2}+3a+2}{a^{2}-2a}$$

Short Solution Steps

Factor the expressions that are not already factored in \frac{a^{2}-4}{a^{2}-4a+4}.

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

Cancel out a-2 in both numerator and denominator.

$$\frac{a+2}{a-2}-\frac{1}{2-a}+\frac{2}{a^{2}-2a}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of a-2 and 2-a is a-2. Multiply \frac{1}{2-a} times \frac{-1}{-1}.

$$\frac{a+2}{a-2}-\frac{-1}{a-2}+\frac{2}{a^{2}-2a}$$

Since \frac{a+2}{a-2} and \frac{-1}{a-2} have the same denominator, subtract them by subtracting their numerators.

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

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

$$\frac{a+2+1}{a-2}+\frac{2}{a^{2}-2a}$$

Combine like terms in a+2+1.

$$\frac{a+3}{a-2}+\frac{2}{a^{2}-2a}$$

Factor a^{2}-2a.

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

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

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

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

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

Do the multiplications in \left(a+3\right)a+2.

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

Expand a\left(a-2\right).

$$\frac{a^{2}+3a+2}{a^{2}-2a}$$