Solve (a+1-4a-5/a-1)div(1/a-1/a^2-a)

$$( a + 1 - \frac { 4 a - 5 } { a - 1 } ) \div ( \frac { 1 } { a } - \frac { 1 } { a ^ { 2 } - a } )$$

Evaluate

$$a\left(a-2\right)$$

Short Solution Steps

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

$$\frac{\frac{\left(a+1\right)\left(a-1\right)}{a-1}-\frac{4a-5}{a-1}}{\frac{1}{a}-\frac{1}{a^{2}-a}}$$

Since \frac{\left(a+1\right)\left(a-1\right)}{a-1} and \frac{4a-5}{a-1} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{\left(a+1\right)\left(a-1\right)-\left(4a-5\right)}{a-1}}{\frac{1}{a}-\frac{1}{a^{2}-a}}$$

Do the multiplications in \left(a+1\right)\left(a-1\right)-\left(4a-5\right).

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

Combine like terms in a^{2}-a-1+a-4a+5.

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

Factor a^{2}-a.

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

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

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

Since \frac{a-1}{a\left(a-1\right)} and \frac{1}{a\left(a-1\right)} have the same denominator, subtract them by subtracting their numerators.

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

Combine like terms in a-1-1.

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

Divide \frac{a^{2}-4a+4}{a-1} by \frac{a-2}{a\left(a-1\right)} by multiplying \frac{a^{2}-4a+4}{a-1} by the reciprocal of \frac{a-2}{a\left(a-1\right)}.

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

Cancel out a-1 in both numerator and denominator.

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

Factor the expressions that are not already factored.

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

Cancel out a-2 in both numerator and denominator.

$$a\left(a-2\right)$$

Expand the expression.

$$a^{2}-2a$$

Expand

$$a^{2}-2a$$

Short Solution Steps

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

$$\frac{\frac{\left(a+1\right)\left(a-1\right)}{a-1}-\frac{4a-5}{a-1}}{\frac{1}{a}-\frac{1}{a^{2}-a}}$$

Since \frac{\left(a+1\right)\left(a-1\right)}{a-1} and \frac{4a-5}{a-1} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{\left(a+1\right)\left(a-1\right)-\left(4a-5\right)}{a-1}}{\frac{1}{a}-\frac{1}{a^{2}-a}}$$

Do the multiplications in \left(a+1\right)\left(a-1\right)-\left(4a-5\right).

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

Combine like terms in a^{2}-a-1+a-4a+5.

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

Factor a^{2}-a.

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

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

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

Since \frac{a-1}{a\left(a-1\right)} and \frac{1}{a\left(a-1\right)} have the same denominator, subtract them by subtracting their numerators.

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

Combine like terms in a-1-1.

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

Divide \frac{a^{2}-4a+4}{a-1} by \frac{a-2}{a\left(a-1\right)} by multiplying \frac{a^{2}-4a+4}{a-1} by the reciprocal of \frac{a-2}{a\left(a-1\right)}.

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

Cancel out a-1 in both numerator and denominator.

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

Factor the expressions that are not already factored.

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

Cancel out a-2 in both numerator and denominator.

$$a\left(a-2\right)$$

Expand the expression.

$$a^{2}-2a$$