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\frac{3}{\left(x-2\right)\left(x+2\right)}+\frac{2}{\left(x-2\right)^{2}}
Factor x^{2}-4.
\frac{3\left(x-2\right)}{\left(x+2\right)\left(x-2\right)^{2}}+\frac{2\left(x+2\right)}{\left(x+2\right)\left(x-2\right)^{2}}
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 \left(x-2\right)^{2} is \left(x+2\right)\left(x-2\right)^{2}. Multiply \frac{3}{\left(x-2\right)\left(x+2\right)} times \frac{x-2}{x-2}. Multiply \frac{2}{\left(x-2\right)^{2}} times \frac{x+2}{x+2}.
\frac{3\left(x-2\right)+2\left(x+2\right)}{\left(x+2\right)\left(x-2\right)^{2}}
Since \frac{3\left(x-2\right)}{\left(x+2\right)\left(x-2\right)^{2}} and \frac{2\left(x+2\right)}{\left(x+2\right)\left(x-2\right)^{2}} have the same denominator, add them by adding their numerators.
\frac{3x-6+2x+4}{\left(x+2\right)\left(x-2\right)^{2}}
Do the multiplications in 3\left(x-2\right)+2\left(x+2\right).
\frac{5x-2}{\left(x+2\right)\left(x-2\right)^{2}}
Combine like terms in 3x-6+2x+4.
\frac{5x-2}{x^{3}-2x^{2}-4x+8}
Expand \left(x+2\right)\left(x-2\right)^{2}.
\frac{3}{\left(x-2\right)\left(x+2\right)}+\frac{2}{\left(x-2\right)^{2}}
Factor x^{2}-4.
\frac{3\left(x-2\right)}{\left(x+2\right)\left(x-2\right)^{2}}+\frac{2\left(x+2\right)}{\left(x+2\right)\left(x-2\right)^{2}}
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 \left(x-2\right)^{2} is \left(x+2\right)\left(x-2\right)^{2}. Multiply \frac{3}{\left(x-2\right)\left(x+2\right)} times \frac{x-2}{x-2}. Multiply \frac{2}{\left(x-2\right)^{2}} times \frac{x+2}{x+2}.
\frac{3\left(x-2\right)+2\left(x+2\right)}{\left(x+2\right)\left(x-2\right)^{2}}
Since \frac{3\left(x-2\right)}{\left(x+2\right)\left(x-2\right)^{2}} and \frac{2\left(x+2\right)}{\left(x+2\right)\left(x-2\right)^{2}} have the same denominator, add them by adding their numerators.
\frac{3x-6+2x+4}{\left(x+2\right)\left(x-2\right)^{2}}
Do the multiplications in 3\left(x-2\right)+2\left(x+2\right).
\frac{5x-2}{\left(x+2\right)\left(x-2\right)^{2}}
Combine like terms in 3x-6+2x+4.
\frac{5x-2}{x^{3}-2x^{2}-4x+8}
Expand \left(x+2\right)\left(x-2\right)^{2}.