Solve a+1/a^2-a-6-a-4/a^2-4a+3+a+4/a^2+a-2

$$\frac { a + 1 } { a ^ { 2 } - a - 6 } - \frac { a - 4 } { a ^ { 2 } - 4 a + 3 } + \frac { a + 4 } { a ^ { 2 } + a - 2 }$$

Evaluate

$$\frac{a^{2}+3a-5}{\left(a-3\right)\left(a-1\right)\left(a+2\right)}$$

Short Solution Steps

Factor a^{2}-a-6. Factor a^{2}-4a+3.

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

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

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

Since \frac{\left(a+1\right)\left(a-1\right)}{\left(a-3\right)\left(a-1\right)\left(a+2\right)} and \frac{\left(a-4\right)\left(a+2\right)}{\left(a-3\right)\left(a-1\right)\left(a+2\right)} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in \left(a+1\right)\left(a-1\right)-\left(a-4\right)\left(a+2\right).

$$\frac{a^{2}-a+a-1-a^{2}-2a+4a+8}{\left(a-3\right)\left(a-1\right)\left(a+2\right)}+\frac{a+4}{a^{2}+a-2}$$

Combine like terms in a^{2}-a+a-1-a^{2}-2a+4a+8.

$$\frac{2a+7}{\left(a-3\right)\left(a-1\right)\left(a+2\right)}+\frac{a+4}{a^{2}+a-2}$$

Factor a^{2}+a-2.

$$\frac{2a+7}{\left(a-3\right)\left(a-1\right)\left(a+2\right)}+\frac{a+4}{\left(a-1\right)\left(a+2\right)}$$

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

$$\frac{2a+7}{\left(a-3\right)\left(a-1\right)\left(a+2\right)}+\frac{\left(a+4\right)\left(a-3\right)}{\left(a-3\right)\left(a-1\right)\left(a+2\right)}$$

Since \frac{2a+7}{\left(a-3\right)\left(a-1\right)\left(a+2\right)} and \frac{\left(a+4\right)\left(a-3\right)}{\left(a-3\right)\left(a-1\right)\left(a+2\right)} have the same denominator, add them by adding their numerators.

$$\frac{2a+7+\left(a+4\right)\left(a-3\right)}{\left(a-3\right)\left(a-1\right)\left(a+2\right)}$$

Do the multiplications in 2a+7+\left(a+4\right)\left(a-3\right).

$$\frac{2a+7+a^{2}-3a+4a-12}{\left(a-3\right)\left(a-1\right)\left(a+2\right)}$$

Combine like terms in 2a+7+a^{2}-3a+4a-12.

$$\frac{3a-5+a^{2}}{\left(a-3\right)\left(a-1\right)\left(a+2\right)}$$

Expand \left(a-3\right)\left(a-1\right)\left(a+2\right).

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

Expand

$$\frac{a^{2}+3a-5}{\left(a-3\right)\left(a-1\right)\left(a+2\right)}$$

Short Solution Steps

Factor a^{2}-a-6. Factor a^{2}-4a+3.

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

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

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

Since \frac{\left(a+1\right)\left(a-1\right)}{\left(a-3\right)\left(a-1\right)\left(a+2\right)} and \frac{\left(a-4\right)\left(a+2\right)}{\left(a-3\right)\left(a-1\right)\left(a+2\right)} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in \left(a+1\right)\left(a-1\right)-\left(a-4\right)\left(a+2\right).

$$\frac{a^{2}-a+a-1-a^{2}-2a+4a+8}{\left(a-3\right)\left(a-1\right)\left(a+2\right)}+\frac{a+4}{a^{2}+a-2}$$

Combine like terms in a^{2}-a+a-1-a^{2}-2a+4a+8.

$$\frac{2a+7}{\left(a-3\right)\left(a-1\right)\left(a+2\right)}+\frac{a+4}{a^{2}+a-2}$$

Factor a^{2}+a-2.

$$\frac{2a+7}{\left(a-3\right)\left(a-1\right)\left(a+2\right)}+\frac{a+4}{\left(a-1\right)\left(a+2\right)}$$

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

$$\frac{2a+7}{\left(a-3\right)\left(a-1\right)\left(a+2\right)}+\frac{\left(a+4\right)\left(a-3\right)}{\left(a-3\right)\left(a-1\right)\left(a+2\right)}$$

Since \frac{2a+7}{\left(a-3\right)\left(a-1\right)\left(a+2\right)} and \frac{\left(a+4\right)\left(a-3\right)}{\left(a-3\right)\left(a-1\right)\left(a+2\right)} have the same denominator, add them by adding their numerators.

$$\frac{2a+7+\left(a+4\right)\left(a-3\right)}{\left(a-3\right)\left(a-1\right)\left(a+2\right)}$$

Do the multiplications in 2a+7+\left(a+4\right)\left(a-3\right).

$$\frac{2a+7+a^{2}-3a+4a-12}{\left(a-3\right)\left(a-1\right)\left(a+2\right)}$$

Combine like terms in 2a+7+a^{2}-3a+4a-12.

$$\frac{3a-5+a^{2}}{\left(a-3\right)\left(a-1\right)\left(a+2\right)}$$

Expand \left(a-3\right)\left(a-1\right)\left(a+2\right).

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