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\frac{3\left(x+4\right)}{\left(x-5\right)\left(x+2\right)\left(x+4\right)}-\frac{4\left(x-5\right)}{\left(x-5\right)\left(x+2\right)\left(x+4\right)}
To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-5\right)\left(x+2\right) and \left(x+2\right)\left(x+4\right) is \left(x-5\right)\left(x+2\right)\left(x+4\right). Multiply \frac{3}{\left(x-5\right)\left(x+2\right)} times \frac{x+4}{x+4}. Multiply \frac{4}{\left(x+2\right)\left(x+4\right)} times \frac{x-5}{x-5}.
\frac{3\left(x+4\right)-4\left(x-5\right)}{\left(x-5\right)\left(x+2\right)\left(x+4\right)}
Since \frac{3\left(x+4\right)}{\left(x-5\right)\left(x+2\right)\left(x+4\right)} and \frac{4\left(x-5\right)}{\left(x-5\right)\left(x+2\right)\left(x+4\right)} have the same denominator, subtract them by subtracting their numerators.
\frac{3x+12-4x+20}{\left(x-5\right)\left(x+2\right)\left(x+4\right)}
Do the multiplications in 3\left(x+4\right)-4\left(x-5\right).
\frac{-x+32}{\left(x-5\right)\left(x+2\right)\left(x+4\right)}
Combine like terms in 3x+12-4x+20.
\frac{-x+32}{x^{3}+x^{2}-22x-40}
Expand \left(x-5\right)\left(x+2\right)\left(x+4\right).
\frac{3\left(x+4\right)}{\left(x-5\right)\left(x+2\right)\left(x+4\right)}-\frac{4\left(x-5\right)}{\left(x-5\right)\left(x+2\right)\left(x+4\right)}
To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-5\right)\left(x+2\right) and \left(x+2\right)\left(x+4\right) is \left(x-5\right)\left(x+2\right)\left(x+4\right). Multiply \frac{3}{\left(x-5\right)\left(x+2\right)} times \frac{x+4}{x+4}. Multiply \frac{4}{\left(x+2\right)\left(x+4\right)} times \frac{x-5}{x-5}.
\frac{3\left(x+4\right)-4\left(x-5\right)}{\left(x-5\right)\left(x+2\right)\left(x+4\right)}
Since \frac{3\left(x+4\right)}{\left(x-5\right)\left(x+2\right)\left(x+4\right)} and \frac{4\left(x-5\right)}{\left(x-5\right)\left(x+2\right)\left(x+4\right)} have the same denominator, subtract them by subtracting their numerators.
\frac{3x+12-4x+20}{\left(x-5\right)\left(x+2\right)\left(x+4\right)}
Do the multiplications in 3\left(x+4\right)-4\left(x-5\right).
\frac{-x+32}{\left(x-5\right)\left(x+2\right)\left(x+4\right)}
Combine like terms in 3x+12-4x+20.
\frac{-x+32}{x^{3}+x^{2}-22x-40}
Expand \left(x-5\right)\left(x+2\right)\left(x+4\right).