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

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

Evaluate

$$\frac{5a^{2}+20a+4}{\left(a-2\right)\left(a+2\right)^{2}}$$

Short Solution Steps

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

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

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

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

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

$$\frac{3a-6+4a+8}{\left(a-2\right)\left(a+2\right)}-\frac{2a}{a^{2}+4a+4}$$

Combine like terms in 3a-6+4a+8.

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

Factor a^{2}+4a+4.

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

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) and \left(a+2\right)^{2} is \left(a-2\right)\left(a+2\right)^{2}. Multiply \frac{7a+2}{\left(a-2\right)\left(a+2\right)} times \frac{a+2}{a+2}. Multiply \frac{2a}{\left(a+2\right)^{2}} times \frac{a-2}{a-2}.

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

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

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

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

$$\frac{7a^{2}+14a+2a+4-2a^{2}+4a}{\left(a-2\right)\left(a+2\right)^{2}}$$

Combine like terms in 7a^{2}+14a+2a+4-2a^{2}+4a.

$$\frac{5a^{2}+20a+4}{\left(a-2\right)\left(a+2\right)^{2}}$$

Expand \left(a-2\right)\left(a+2\right)^{2}.

$$\frac{5a^{2}+20a+4}{a^{3}+2a^{2}-4a-8}$$

Differentiate w.r.t. a

$$-\frac{\left(a+6\right)\left(5a^{2}+12\right)}{\left(a-2\right)^{2}\left(a+2\right)^{3}}$$