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

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

Evaluate

$$-\frac{2a^{3}+5a^{2}+66a+56}{\left(5a+4\right)\left(a^{2}-4\right)}$$

Short Solution Steps

Combine a and 4a to get 5a.

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

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}.

$$\frac{3\left(a-2\right)}{\left(a-2\right)\left(a+2\right)}-\frac{4\left(a+2\right)}{\left(a-2\right)\left(a+2\right)}-\frac{2a}{5a+4}$$

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, subtract them by subtracting their numerators.

$$\frac{3\left(a-2\right)-4\left(a+2\right)}{\left(a-2\right)\left(a+2\right)}-\frac{2a}{5a+4}$$

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

$$\frac{3a-6-4a-8}{\left(a-2\right)\left(a+2\right)}-\frac{2a}{5a+4}$$

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

$$\frac{-a-14}{\left(a-2\right)\left(a+2\right)}-\frac{2a}{5a+4}$$

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 5a+4 is \left(a-2\right)\left(a+2\right)\left(5a+4\right). Multiply \frac{-a-14}{\left(a-2\right)\left(a+2\right)} times \frac{5a+4}{5a+4}. Multiply \frac{2a}{5a+4} times \frac{\left(a-2\right)\left(a+2\right)}{\left(a-2\right)\left(a+2\right)}.

$$\frac{\left(-a-14\right)\left(5a+4\right)}{\left(a-2\right)\left(a+2\right)\left(5a+4\right)}-\frac{2a\left(a-2\right)\left(a+2\right)}{\left(a-2\right)\left(a+2\right)\left(5a+4\right)}$$

Since \frac{\left(-a-14\right)\left(5a+4\right)}{\left(a-2\right)\left(a+2\right)\left(5a+4\right)} and \frac{2a\left(a-2\right)\left(a+2\right)}{\left(a-2\right)\left(a+2\right)\left(5a+4\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(-a-14\right)\left(5a+4\right)-2a\left(a-2\right)\left(a+2\right)}{\left(a-2\right)\left(a+2\right)\left(5a+4\right)}$$

Do the multiplications in \left(-a-14\right)\left(5a+4\right)-2a\left(a-2\right)\left(a+2\right).

$$\frac{-5a^{2}-4a-70a-56-2a^{3}-4a^{2}+4a^{2}+8a}{\left(a-2\right)\left(a+2\right)\left(5a+4\right)}$$

Combine like terms in -5a^{2}-4a-70a-56-2a^{3}-4a^{2}+4a^{2}+8a.

$$\frac{-5a^{2}-66a-56-2a^{3}}{\left(a-2\right)\left(a+2\right)\left(5a+4\right)}$$

Expand \left(a-2\right)\left(a+2\right)\left(5a+4\right).

$$\frac{-5a^{2}-66a-56-2a^{3}}{5a^{3}+4a^{2}-20a-16}$$

Differentiate w.r.t. a

$$\frac{17a^{4}+740a^{3}+1300a^{2}+608a-64}{\left(\left(5a+4\right)\left(a^{2}-4\right)\right)^{2}}$$