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

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

Evaluate

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

Short Solution Steps

Factor a^{2}+a.

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

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

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

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

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

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

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

Combine like terms in ba-2b-ba+a.

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

Expand a\left(a-2\right)\left(a+1\right).

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

Expand

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

Short Solution Steps

Factor a^{2}+a.

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

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

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

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

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

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

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

Combine like terms in ba-2b-ba+a.

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

Expand a\left(a-2\right)\left(a+1\right).

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