Solve 3c-5/c^2-4+4/2-c-4/c+2+1/1

$$\frac{ 3c-5 }{ { c }^{ 2 } -4 } + \frac{ 4 }{ 2-c } - \frac{ 4 }{ c+2 } + \frac{ 1 }{ 1 }$$

Evaluate

$$\frac{9+5c-c^{2}}{4-c^{2}}$$

Short Solution Steps

Divide 1 by 1 to get 1.

$$\frac{3c-5}{c^{2}-4}+\frac{4}{2-c}-\frac{4}{c+2}+1$$

Factor c^{2}-4.

$$\frac{3c-5}{\left(c-2\right)\left(c+2\right)}+\frac{4}{2-c}-\frac{4}{c+2}+1$$

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

$$\frac{3c-5}{\left(c-2\right)\left(c+2\right)}+\frac{4\left(-1\right)\left(c+2\right)}{\left(c-2\right)\left(c+2\right)}-\frac{4}{c+2}+1$$

Since \frac{3c-5}{\left(c-2\right)\left(c+2\right)} and \frac{4\left(-1\right)\left(c+2\right)}{\left(c-2\right)\left(c+2\right)} have the same denominator, add them by adding their numerators.

$$\frac{3c-5+4\left(-1\right)\left(c+2\right)}{\left(c-2\right)\left(c+2\right)}-\frac{4}{c+2}+1$$

Do the multiplications in 3c-5+4\left(-1\right)\left(c+2\right).

$$\frac{3c-5-4c-8}{\left(c-2\right)\left(c+2\right)}-\frac{4}{c+2}+1$$

Combine like terms in 3c-5-4c-8.

$$\frac{-c-13}{\left(c-2\right)\left(c+2\right)}-\frac{4}{c+2}+1$$

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

$$\frac{-c-13}{\left(c-2\right)\left(c+2\right)}-\frac{4\left(c-2\right)}{\left(c-2\right)\left(c+2\right)}+1$$

Since \frac{-c-13}{\left(c-2\right)\left(c+2\right)} and \frac{4\left(c-2\right)}{\left(c-2\right)\left(c+2\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{-c-13-4\left(c-2\right)}{\left(c-2\right)\left(c+2\right)}+1$$

Do the multiplications in -c-13-4\left(c-2\right).

$$\frac{-c-13-4c+8}{\left(c-2\right)\left(c+2\right)}+1$$

Combine like terms in -c-13-4c+8.

$$\frac{-5c-5}{\left(c-2\right)\left(c+2\right)}+1$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{\left(c-2\right)\left(c+2\right)}{\left(c-2\right)\left(c+2\right)}.

$$\frac{-5c-5}{\left(c-2\right)\left(c+2\right)}+\frac{\left(c-2\right)\left(c+2\right)}{\left(c-2\right)\left(c+2\right)}$$

Since \frac{-5c-5}{\left(c-2\right)\left(c+2\right)} and \frac{\left(c-2\right)\left(c+2\right)}{\left(c-2\right)\left(c+2\right)} have the same denominator, add them by adding their numerators.

$$\frac{-5c-5+\left(c-2\right)\left(c+2\right)}{\left(c-2\right)\left(c+2\right)}$$

Do the multiplications in -5c-5+\left(c-2\right)\left(c+2\right).

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

Combine like terms in -5c-5+c^{2}+2c-2c-4.

$$\frac{-5c-9+c^{2}}{\left(c-2\right)\left(c+2\right)}$$

Expand \left(c-2\right)\left(c+2\right).

$$\frac{-5c-9+c^{2}}{c^{2}-4}$$

Expand

$$\frac{c^{2}-5c-9}{c^{2}-4}$$

Short Solution Steps

Divide 1 by 1 to get 1.

$$\frac{3c-5}{c^{2}-4}+\frac{4}{2-c}-\frac{4}{c+2}+1$$

Factor c^{2}-4.

$$\frac{3c-5}{\left(c-2\right)\left(c+2\right)}+\frac{4}{2-c}-\frac{4}{c+2}+1$$

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

$$\frac{3c-5}{\left(c-2\right)\left(c+2\right)}+\frac{4\left(-1\right)\left(c+2\right)}{\left(c-2\right)\left(c+2\right)}-\frac{4}{c+2}+1$$

Since \frac{3c-5}{\left(c-2\right)\left(c+2\right)} and \frac{4\left(-1\right)\left(c+2\right)}{\left(c-2\right)\left(c+2\right)} have the same denominator, add them by adding their numerators.

$$\frac{3c-5+4\left(-1\right)\left(c+2\right)}{\left(c-2\right)\left(c+2\right)}-\frac{4}{c+2}+1$$

Do the multiplications in 3c-5+4\left(-1\right)\left(c+2\right).

$$\frac{3c-5-4c-8}{\left(c-2\right)\left(c+2\right)}-\frac{4}{c+2}+1$$

Combine like terms in 3c-5-4c-8.

$$\frac{-c-13}{\left(c-2\right)\left(c+2\right)}-\frac{4}{c+2}+1$$

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

$$\frac{-c-13}{\left(c-2\right)\left(c+2\right)}-\frac{4\left(c-2\right)}{\left(c-2\right)\left(c+2\right)}+1$$

Since \frac{-c-13}{\left(c-2\right)\left(c+2\right)} and \frac{4\left(c-2\right)}{\left(c-2\right)\left(c+2\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{-c-13-4\left(c-2\right)}{\left(c-2\right)\left(c+2\right)}+1$$

Do the multiplications in -c-13-4\left(c-2\right).

$$\frac{-c-13-4c+8}{\left(c-2\right)\left(c+2\right)}+1$$

Combine like terms in -c-13-4c+8.

$$\frac{-5c-5}{\left(c-2\right)\left(c+2\right)}+1$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{\left(c-2\right)\left(c+2\right)}{\left(c-2\right)\left(c+2\right)}.

$$\frac{-5c-5}{\left(c-2\right)\left(c+2\right)}+\frac{\left(c-2\right)\left(c+2\right)}{\left(c-2\right)\left(c+2\right)}$$

Since \frac{-5c-5}{\left(c-2\right)\left(c+2\right)} and \frac{\left(c-2\right)\left(c+2\right)}{\left(c-2\right)\left(c+2\right)} have the same denominator, add them by adding their numerators.

$$\frac{-5c-5+\left(c-2\right)\left(c+2\right)}{\left(c-2\right)\left(c+2\right)}$$

Do the multiplications in -5c-5+\left(c-2\right)\left(c+2\right).

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

Combine like terms in -5c-5+c^{2}+2c-2c-4.

$$\frac{-5c-9+c^{2}}{\left(c-2\right)\left(c+2\right)}$$

Expand \left(c-2\right)\left(c+2\right).

$$\frac{-5c-9+c^{2}}{c^{2}-4}$$