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\frac{x-3}{\left(x-2\right)\left(x+2\right)}-\frac{5}{x+2}
Factor x^{2}-4.
\frac{x-3}{\left(x-2\right)\left(x+2\right)}-\frac{5\left(x-2\right)}{\left(x-2\right)\left(x+2\right)}
To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-2\right)\left(x+2\right) and x+2 is \left(x-2\right)\left(x+2\right). Multiply \frac{5}{x+2} times \frac{x-2}{x-2}.
\frac{x-3-5\left(x-2\right)}{\left(x-2\right)\left(x+2\right)}
Since \frac{x-3}{\left(x-2\right)\left(x+2\right)} and \frac{5\left(x-2\right)}{\left(x-2\right)\left(x+2\right)} have the same denominator, subtract them by subtracting their numerators.
\frac{x-3-5x+10}{\left(x-2\right)\left(x+2\right)}
Do the multiplications in x-3-5\left(x-2\right).
\frac{-4x+7}{\left(x-2\right)\left(x+2\right)}
Combine like terms in x-3-5x+10.
\frac{-4x+7}{x^{2}-4}
Expand \left(x-2\right)\left(x+2\right).
\frac{x-3}{\left(x-2\right)\left(x+2\right)}-\frac{5}{x+2}
Factor x^{2}-4.
\frac{x-3}{\left(x-2\right)\left(x+2\right)}-\frac{5\left(x-2\right)}{\left(x-2\right)\left(x+2\right)}
To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-2\right)\left(x+2\right) and x+2 is \left(x-2\right)\left(x+2\right). Multiply \frac{5}{x+2} times \frac{x-2}{x-2}.
\frac{x-3-5\left(x-2\right)}{\left(x-2\right)\left(x+2\right)}
Since \frac{x-3}{\left(x-2\right)\left(x+2\right)} and \frac{5\left(x-2\right)}{\left(x-2\right)\left(x+2\right)} have the same denominator, subtract them by subtracting their numerators.
\frac{x-3-5x+10}{\left(x-2\right)\left(x+2\right)}
Do the multiplications in x-3-5\left(x-2\right).
\frac{-4x+7}{\left(x-2\right)\left(x+2\right)}
Combine like terms in x-3-5x+10.
\frac{-4x+7}{x^{2}-4}
Expand \left(x-2\right)\left(x+2\right).