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

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

Evaluate

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

Short Solution Steps

Combine -4a and a to get -3a.

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

Factor a^{2}-3a.

$$\frac{a^{2}-4}{a\left(a-3\right)}-\frac{1}{2-a}+\frac{2}{a^{2}-2a}$$

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

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

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

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

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

$$\frac{-a^{3}+2a^{2}+4a-8-a^{2}+3a}{a\left(a-3\right)\left(-a+2\right)}+\frac{2}{a^{2}-2a}$$

Combine like terms in -a^{3}+2a^{2}+4a-8-a^{2}+3a.

$$\frac{-a^{3}+a^{2}+7a-8}{a\left(a-3\right)\left(-a+2\right)}+\frac{2}{a^{2}-2a}$$

Factor a^{2}-2a.

$$\frac{-a^{3}+a^{2}+7a-8}{a\left(a-3\right)\left(-a+2\right)}+\frac{2}{a\left(a-2\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of a\left(a-3\right)\left(-a+2\right) and a\left(a-2\right) is a\left(a-3\right)\left(a-2\right). Multiply \frac{-a^{3}+a^{2}+7a-8}{a\left(a-3\right)\left(-a+2\right)} times \frac{-1}{-1}. Multiply \frac{2}{a\left(a-2\right)} times \frac{a-3}{a-3}.

$$\frac{-\left(-a^{3}+a^{2}+7a-8\right)}{a\left(a-3\right)\left(a-2\right)}+\frac{2\left(a-3\right)}{a\left(a-3\right)\left(a-2\right)}$$

Since \frac{-\left(-a^{3}+a^{2}+7a-8\right)}{a\left(a-3\right)\left(a-2\right)} and \frac{2\left(a-3\right)}{a\left(a-3\right)\left(a-2\right)} have the same denominator, add them by adding their numerators.

$$\frac{-\left(-a^{3}+a^{2}+7a-8\right)+2\left(a-3\right)}{a\left(a-3\right)\left(a-2\right)}$$

Do the multiplications in -\left(-a^{3}+a^{2}+7a-8\right)+2\left(a-3\right).

$$\frac{a^{3}-a^{2}-7a+8+2a-6}{a\left(a-3\right)\left(a-2\right)}$$

Combine like terms in a^{3}-a^{2}-7a+8+2a-6.

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

Expand a\left(a-3\right)\left(a-2\right).

$$\frac{a^{3}-a^{2}-5a+2}{a^{3}-5a^{2}+6a}$$

Expand

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

Short Solution Steps

Combine -4a and a to get -3a.

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

Factor a^{2}-3a.

$$\frac{a^{2}-4}{a\left(a-3\right)}-\frac{1}{2-a}+\frac{2}{a^{2}-2a}$$

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

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

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

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

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

$$\frac{-a^{3}+2a^{2}+4a-8-a^{2}+3a}{a\left(a-3\right)\left(-a+2\right)}+\frac{2}{a^{2}-2a}$$

Combine like terms in -a^{3}+2a^{2}+4a-8-a^{2}+3a.

$$\frac{-a^{3}+a^{2}+7a-8}{a\left(a-3\right)\left(-a+2\right)}+\frac{2}{a^{2}-2a}$$

Factor a^{2}-2a.

$$\frac{-a^{3}+a^{2}+7a-8}{a\left(a-3\right)\left(-a+2\right)}+\frac{2}{a\left(a-2\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of a\left(a-3\right)\left(-a+2\right) and a\left(a-2\right) is a\left(a-3\right)\left(a-2\right). Multiply \frac{-a^{3}+a^{2}+7a-8}{a\left(a-3\right)\left(-a+2\right)} times \frac{-1}{-1}. Multiply \frac{2}{a\left(a-2\right)} times \frac{a-3}{a-3}.

$$\frac{-\left(-a^{3}+a^{2}+7a-8\right)}{a\left(a-3\right)\left(a-2\right)}+\frac{2\left(a-3\right)}{a\left(a-3\right)\left(a-2\right)}$$

Since \frac{-\left(-a^{3}+a^{2}+7a-8\right)}{a\left(a-3\right)\left(a-2\right)} and \frac{2\left(a-3\right)}{a\left(a-3\right)\left(a-2\right)} have the same denominator, add them by adding their numerators.

$$\frac{-\left(-a^{3}+a^{2}+7a-8\right)+2\left(a-3\right)}{a\left(a-3\right)\left(a-2\right)}$$

Do the multiplications in -\left(-a^{3}+a^{2}+7a-8\right)+2\left(a-3\right).

$$\frac{a^{3}-a^{2}-7a+8+2a-6}{a\left(a-3\right)\left(a-2\right)}$$

Combine like terms in a^{3}-a^{2}-7a+8+2a-6.

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

Expand a\left(a-3\right)\left(a-2\right).

$$\frac{a^{3}-a^{2}-5a+2}{a^{3}-5a^{2}+6a}$$