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\frac{x+5}{\left(x-3\right)\left(x+4\right)}+\frac{x+4}{\left(x-3\right)\left(x+5\right)}+\frac{x-3}{x^{2}+9x+20}
Factor x^{2}+x-12. Factor x^{2}+2x-15.
\frac{\left(x+5\right)\left(x+5\right)}{\left(x-3\right)\left(x+4\right)\left(x+5\right)}+\frac{\left(x+4\right)\left(x+4\right)}{\left(x-3\right)\left(x+4\right)\left(x+5\right)}+\frac{x-3}{x^{2}+9x+20}
To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-3\right)\left(x+4\right) and \left(x-3\right)\left(x+5\right) is \left(x-3\right)\left(x+4\right)\left(x+5\right). Multiply \frac{x+5}{\left(x-3\right)\left(x+4\right)} times \frac{x+5}{x+5}. Multiply \frac{x+4}{\left(x-3\right)\left(x+5\right)} times \frac{x+4}{x+4}.
\frac{\left(x+5\right)\left(x+5\right)+\left(x+4\right)\left(x+4\right)}{\left(x-3\right)\left(x+4\right)\left(x+5\right)}+\frac{x-3}{x^{2}+9x+20}
Since \frac{\left(x+5\right)\left(x+5\right)}{\left(x-3\right)\left(x+4\right)\left(x+5\right)} and \frac{\left(x+4\right)\left(x+4\right)}{\left(x-3\right)\left(x+4\right)\left(x+5\right)} have the same denominator, add them by adding their numerators.
\frac{x^{2}+5x+5x+25+x^{2}+4x+4x+16}{\left(x-3\right)\left(x+4\right)\left(x+5\right)}+\frac{x-3}{x^{2}+9x+20}
Do the multiplications in \left(x+5\right)\left(x+5\right)+\left(x+4\right)\left(x+4\right).
\frac{2x^{2}+18x+41}{\left(x-3\right)\left(x+4\right)\left(x+5\right)}+\frac{x-3}{x^{2}+9x+20}
Combine like terms in x^{2}+5x+5x+25+x^{2}+4x+4x+16.
\frac{2x^{2}+18x+41}{\left(x-3\right)\left(x+4\right)\left(x+5\right)}+\frac{x-3}{\left(x+4\right)\left(x+5\right)}
Factor x^{2}+9x+20.
\frac{2x^{2}+18x+41}{\left(x-3\right)\left(x+4\right)\left(x+5\right)}+\frac{\left(x-3\right)\left(x-3\right)}{\left(x-3\right)\left(x+4\right)\left(x+5\right)}
To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-3\right)\left(x+4\right)\left(x+5\right) and \left(x+4\right)\left(x+5\right) is \left(x-3\right)\left(x+4\right)\left(x+5\right). Multiply \frac{x-3}{\left(x+4\right)\left(x+5\right)} times \frac{x-3}{x-3}.
\frac{2x^{2}+18x+41+\left(x-3\right)\left(x-3\right)}{\left(x-3\right)\left(x+4\right)\left(x+5\right)}
Since \frac{2x^{2}+18x+41}{\left(x-3\right)\left(x+4\right)\left(x+5\right)} and \frac{\left(x-3\right)\left(x-3\right)}{\left(x-3\right)\left(x+4\right)\left(x+5\right)} have the same denominator, add them by adding their numerators.
\frac{2x^{2}+18x+41+x^{2}-3x-3x+9}{\left(x-3\right)\left(x+4\right)\left(x+5\right)}
Do the multiplications in 2x^{2}+18x+41+\left(x-3\right)\left(x-3\right).
\frac{3x^{2}+12x+50}{\left(x-3\right)\left(x+4\right)\left(x+5\right)}
Combine like terms in 2x^{2}+18x+41+x^{2}-3x-3x+9.
\frac{3x^{2}+12x+50}{x^{3}+6x^{2}-7x-60}
Expand \left(x-3\right)\left(x+4\right)\left(x+5\right).
\frac{x+5}{\left(x-3\right)\left(x+4\right)}+\frac{x+4}{\left(x-3\right)\left(x+5\right)}+\frac{x-3}{x^{2}+9x+20}
Factor x^{2}+x-12. Factor x^{2}+2x-15.
\frac{\left(x+5\right)\left(x+5\right)}{\left(x-3\right)\left(x+4\right)\left(x+5\right)}+\frac{\left(x+4\right)\left(x+4\right)}{\left(x-3\right)\left(x+4\right)\left(x+5\right)}+\frac{x-3}{x^{2}+9x+20}
To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-3\right)\left(x+4\right) and \left(x-3\right)\left(x+5\right) is \left(x-3\right)\left(x+4\right)\left(x+5\right). Multiply \frac{x+5}{\left(x-3\right)\left(x+4\right)} times \frac{x+5}{x+5}. Multiply \frac{x+4}{\left(x-3\right)\left(x+5\right)} times \frac{x+4}{x+4}.
\frac{\left(x+5\right)\left(x+5\right)+\left(x+4\right)\left(x+4\right)}{\left(x-3\right)\left(x+4\right)\left(x+5\right)}+\frac{x-3}{x^{2}+9x+20}
Since \frac{\left(x+5\right)\left(x+5\right)}{\left(x-3\right)\left(x+4\right)\left(x+5\right)} and \frac{\left(x+4\right)\left(x+4\right)}{\left(x-3\right)\left(x+4\right)\left(x+5\right)} have the same denominator, add them by adding their numerators.
\frac{x^{2}+5x+5x+25+x^{2}+4x+4x+16}{\left(x-3\right)\left(x+4\right)\left(x+5\right)}+\frac{x-3}{x^{2}+9x+20}
Do the multiplications in \left(x+5\right)\left(x+5\right)+\left(x+4\right)\left(x+4\right).
\frac{2x^{2}+18x+41}{\left(x-3\right)\left(x+4\right)\left(x+5\right)}+\frac{x-3}{x^{2}+9x+20}
Combine like terms in x^{2}+5x+5x+25+x^{2}+4x+4x+16.
\frac{2x^{2}+18x+41}{\left(x-3\right)\left(x+4\right)\left(x+5\right)}+\frac{x-3}{\left(x+4\right)\left(x+5\right)}
Factor x^{2}+9x+20.
\frac{2x^{2}+18x+41}{\left(x-3\right)\left(x+4\right)\left(x+5\right)}+\frac{\left(x-3\right)\left(x-3\right)}{\left(x-3\right)\left(x+4\right)\left(x+5\right)}
To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-3\right)\left(x+4\right)\left(x+5\right) and \left(x+4\right)\left(x+5\right) is \left(x-3\right)\left(x+4\right)\left(x+5\right). Multiply \frac{x-3}{\left(x+4\right)\left(x+5\right)} times \frac{x-3}{x-3}.
\frac{2x^{2}+18x+41+\left(x-3\right)\left(x-3\right)}{\left(x-3\right)\left(x+4\right)\left(x+5\right)}
Since \frac{2x^{2}+18x+41}{\left(x-3\right)\left(x+4\right)\left(x+5\right)} and \frac{\left(x-3\right)\left(x-3\right)}{\left(x-3\right)\left(x+4\right)\left(x+5\right)} have the same denominator, add them by adding their numerators.
\frac{2x^{2}+18x+41+x^{2}-3x-3x+9}{\left(x-3\right)\left(x+4\right)\left(x+5\right)}
Do the multiplications in 2x^{2}+18x+41+\left(x-3\right)\left(x-3\right).
\frac{3x^{2}+12x+50}{\left(x-3\right)\left(x+4\right)\left(x+5\right)}
Combine like terms in 2x^{2}+18x+41+x^{2}-3x-3x+9.
\frac{3x^{2}+12x+50}{x^{3}+6x^{2}-7x-60}
Expand \left(x-3\right)\left(x+4\right)\left(x+5\right).