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\frac{2\left(x+1\right)}{\left(x+1\right)\left(x+3\right)}-\frac{x-1}{x^{2}+5x+6}
Factor the expressions that are not already factored in \frac{2x+2}{x^{2}+4x+3}.
\frac{2}{x+3}-\frac{x-1}{x^{2}+5x+6}
Cancel out x+1 in both numerator and denominator.
\frac{2}{x+3}-\frac{x-1}{\left(x+2\right)\left(x+3\right)}
Factor x^{2}+5x+6.
\frac{2\left(x+2\right)}{\left(x+2\right)\left(x+3\right)}-\frac{x-1}{\left(x+2\right)\left(x+3\right)}
To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x+3 and \left(x+2\right)\left(x+3\right) is \left(x+2\right)\left(x+3\right). Multiply \frac{2}{x+3} times \frac{x+2}{x+2}.
\frac{2\left(x+2\right)-\left(x-1\right)}{\left(x+2\right)\left(x+3\right)}
Since \frac{2\left(x+2\right)}{\left(x+2\right)\left(x+3\right)} and \frac{x-1}{\left(x+2\right)\left(x+3\right)} have the same denominator, subtract them by subtracting their numerators.
\frac{2x+4-x+1}{\left(x+2\right)\left(x+3\right)}
Do the multiplications in 2\left(x+2\right)-\left(x-1\right).
\frac{x+5}{\left(x+2\right)\left(x+3\right)}
Combine like terms in 2x+4-x+1.
\frac{x+5}{x^{2}+5x+6}
Expand \left(x+2\right)\left(x+3\right).
\frac{2\left(x+1\right)}{\left(x+1\right)\left(x+3\right)}-\frac{x-1}{x^{2}+5x+6}
Factor the expressions that are not already factored in \frac{2x+2}{x^{2}+4x+3}.
\frac{2}{x+3}-\frac{x-1}{x^{2}+5x+6}
Cancel out x+1 in both numerator and denominator.
\frac{2}{x+3}-\frac{x-1}{\left(x+2\right)\left(x+3\right)}
Factor x^{2}+5x+6.
\frac{2\left(x+2\right)}{\left(x+2\right)\left(x+3\right)}-\frac{x-1}{\left(x+2\right)\left(x+3\right)}
To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x+3 and \left(x+2\right)\left(x+3\right) is \left(x+2\right)\left(x+3\right). Multiply \frac{2}{x+3} times \frac{x+2}{x+2}.
\frac{2\left(x+2\right)-\left(x-1\right)}{\left(x+2\right)\left(x+3\right)}
Since \frac{2\left(x+2\right)}{\left(x+2\right)\left(x+3\right)} and \frac{x-1}{\left(x+2\right)\left(x+3\right)} have the same denominator, subtract them by subtracting their numerators.
\frac{2x+4-x+1}{\left(x+2\right)\left(x+3\right)}
Do the multiplications in 2\left(x+2\right)-\left(x-1\right).
\frac{x+5}{\left(x+2\right)\left(x+3\right)}
Combine like terms in 2x+4-x+1.
\frac{x+5}{x^{2}+5x+6}
Expand \left(x+2\right)\left(x+3\right).