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

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

Evaluate

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Short Solution Steps

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

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

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

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

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

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

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

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

Factor a^{2}-4.

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

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) and \left(a-2\right)\left(a+2\right) is \left(a-2\right)\left(a+2\right). Multiply \frac{4+3a^{2}}{\left(a+2\right)\left(-a+2\right)} times \frac{-1}{-1}.

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

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

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

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

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

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

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

Factor the expressions that are not already factored in \frac{-4+a^{2}}{\left(a-2\right)\left(a+2\right)}.

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

Cancel out \left(a-2\right)\left(a+2\right) in both numerator and denominator.

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Factor

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