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

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

Evaluate

$$\frac{16+2a-a^{2}}{\left(a+2\right)\left(a-2\right)^{2}}$$

Short Solution Steps

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

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

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

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

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

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

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

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

Combine like terms in 4a+8-2a+4.

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

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)^{2} and a-2 is \left(a+2\right)\left(a-2\right)^{2}. Multiply \frac{1}{a-2} times \frac{\left(a-2\right)\left(a+2\right)}{\left(a-2\right)\left(a+2\right)}.

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

Since \frac{2a+12}{\left(a+2\right)\left(a-2\right)^{2}} and \frac{\left(a-2\right)\left(a+2\right)}{\left(a+2\right)\left(a-2\right)^{2}} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in 2a+12-\left(a-2\right)\left(a+2\right).

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

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

$$\frac{2a+16-a^{2}}{\left(a+2\right)\left(a-2\right)^{2}}$$

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

$$\frac{2a+16-a^{2}}{a^{3}-2a^{2}-4a+8}$$

Differentiate w.r.t. a

$$\frac{a^{3}-2a^{2}-44a-40}{\left(a+2\right)^{2}\left(a-2\right)^{3}}$$