Solve 2/a+3-a+1/a^2-4-a+2/(a-2)(a+3)=

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

Evaluate

$$-\frac{8a+15}{\left(a+3\right)\left(a^{2}-4\right)}$$

Short Solution Steps

Factor a^{2}-4.

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

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

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

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

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

Do the multiplications in 2\left(a-2\right)\left(a+2\right)-\left(a+1\right)\left(a+3\right).

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

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

$$\frac{a^{2}-4a-11}{\left(a-2\right)\left(a+2\right)\left(a+3\right)}-\frac{a+2}{\left(a-2\right)\left(a+3\right)}$$

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

$$\frac{a^{2}-4a-11}{\left(a-2\right)\left(a+2\right)\left(a+3\right)}-\frac{\left(a+2\right)\left(a+2\right)}{\left(a-2\right)\left(a+2\right)\left(a+3\right)}$$

Since \frac{a^{2}-4a-11}{\left(a-2\right)\left(a+2\right)\left(a+3\right)} and \frac{\left(a+2\right)\left(a+2\right)}{\left(a-2\right)\left(a+2\right)\left(a+3\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{a^{2}-4a-11-\left(a+2\right)\left(a+2\right)}{\left(a-2\right)\left(a+2\right)\left(a+3\right)}$$

Do the multiplications in a^{2}-4a-11-\left(a+2\right)\left(a+2\right).

$$\frac{a^{2}-4a-11-a^{2}-2a-2a-4}{\left(a-2\right)\left(a+2\right)\left(a+3\right)}$$

Combine like terms in a^{2}-4a-11-a^{2}-2a-2a-4.

$$\frac{-8a-15}{\left(a-2\right)\left(a+2\right)\left(a+3\right)}$$

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

$$\frac{-8a-15}{a^{3}+3a^{2}-4a-12}$$

Expand

$$-\frac{8a+15}{\left(a+3\right)\left(a^{2}-4\right)}$$

Short Solution Steps

Factor a^{2}-4.

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

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

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

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

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

Do the multiplications in 2\left(a-2\right)\left(a+2\right)-\left(a+1\right)\left(a+3\right).

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

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

$$\frac{a^{2}-4a-11}{\left(a-2\right)\left(a+2\right)\left(a+3\right)}-\frac{a+2}{\left(a-2\right)\left(a+3\right)}$$

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

$$\frac{a^{2}-4a-11}{\left(a-2\right)\left(a+2\right)\left(a+3\right)}-\frac{\left(a+2\right)\left(a+2\right)}{\left(a-2\right)\left(a+2\right)\left(a+3\right)}$$

Since \frac{a^{2}-4a-11}{\left(a-2\right)\left(a+2\right)\left(a+3\right)} and \frac{\left(a+2\right)\left(a+2\right)}{\left(a-2\right)\left(a+2\right)\left(a+3\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{a^{2}-4a-11-\left(a+2\right)\left(a+2\right)}{\left(a-2\right)\left(a+2\right)\left(a+3\right)}$$

Do the multiplications in a^{2}-4a-11-\left(a+2\right)\left(a+2\right).

$$\frac{a^{2}-4a-11-a^{2}-2a-2a-4}{\left(a-2\right)\left(a+2\right)\left(a+3\right)}$$

Combine like terms in a^{2}-4a-11-a^{2}-2a-2a-4.

$$\frac{-8a-15}{\left(a-2\right)\left(a+2\right)\left(a+3\right)}$$

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

$$\frac{-8a-15}{a^{3}+3a^{2}-4a-12}$$