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\frac{2\left(y-3\right)}{\left(y-3\right)\left(y+3\right)}-\frac{y}{y-1}+\frac{y^{2}+2}{y^{2}+2y-3}
Factor the expressions that are not already factored in \frac{2y-6}{y^{2}-9}.
\frac{2}{y+3}-\frac{y}{y-1}+\frac{y^{2}+2}{y^{2}+2y-3}
Cancel out y-3 in both numerator and denominator.
\frac{2\left(y-1\right)}{\left(y-1\right)\left(y+3\right)}-\frac{y\left(y+3\right)}{\left(y-1\right)\left(y+3\right)}+\frac{y^{2}+2}{y^{2}+2y-3}
To add or subtract expressions, expand them to make their denominators the same. Least common multiple of y+3 and y-1 is \left(y-1\right)\left(y+3\right). Multiply \frac{2}{y+3} times \frac{y-1}{y-1}. Multiply \frac{y}{y-1} times \frac{y+3}{y+3}.
\frac{2\left(y-1\right)-y\left(y+3\right)}{\left(y-1\right)\left(y+3\right)}+\frac{y^{2}+2}{y^{2}+2y-3}
Since \frac{2\left(y-1\right)}{\left(y-1\right)\left(y+3\right)} and \frac{y\left(y+3\right)}{\left(y-1\right)\left(y+3\right)} have the same denominator, subtract them by subtracting their numerators.
\frac{2y-2-y^{2}-3y}{\left(y-1\right)\left(y+3\right)}+\frac{y^{2}+2}{y^{2}+2y-3}
Do the multiplications in 2\left(y-1\right)-y\left(y+3\right).
\frac{-y-2-y^{2}}{\left(y-1\right)\left(y+3\right)}+\frac{y^{2}+2}{y^{2}+2y-3}
Combine like terms in 2y-2-y^{2}-3y.
\frac{-y-2-y^{2}}{\left(y-1\right)\left(y+3\right)}+\frac{y^{2}+2}{\left(y-1\right)\left(y+3\right)}
Factor y^{2}+2y-3.
\frac{-y-2-y^{2}+y^{2}+2}{\left(y-1\right)\left(y+3\right)}
Since \frac{-y-2-y^{2}}{\left(y-1\right)\left(y+3\right)} and \frac{y^{2}+2}{\left(y-1\right)\left(y+3\right)} have the same denominator, add them by adding their numerators.
\frac{-y}{\left(y-1\right)\left(y+3\right)}
Combine like terms in -y-2-y^{2}+y^{2}+2.
\frac{-y}{y^{2}+2y-3}
Expand \left(y-1\right)\left(y+3\right).
\frac{2\left(y-3\right)}{\left(y-3\right)\left(y+3\right)}-\frac{y}{y-1}+\frac{y^{2}+2}{y^{2}+2y-3}
Factor the expressions that are not already factored in \frac{2y-6}{y^{2}-9}.
\frac{2}{y+3}-\frac{y}{y-1}+\frac{y^{2}+2}{y^{2}+2y-3}
Cancel out y-3 in both numerator and denominator.
\frac{2\left(y-1\right)}{\left(y-1\right)\left(y+3\right)}-\frac{y\left(y+3\right)}{\left(y-1\right)\left(y+3\right)}+\frac{y^{2}+2}{y^{2}+2y-3}
To add or subtract expressions, expand them to make their denominators the same. Least common multiple of y+3 and y-1 is \left(y-1\right)\left(y+3\right). Multiply \frac{2}{y+3} times \frac{y-1}{y-1}. Multiply \frac{y}{y-1} times \frac{y+3}{y+3}.
\frac{2\left(y-1\right)-y\left(y+3\right)}{\left(y-1\right)\left(y+3\right)}+\frac{y^{2}+2}{y^{2}+2y-3}
Since \frac{2\left(y-1\right)}{\left(y-1\right)\left(y+3\right)} and \frac{y\left(y+3\right)}{\left(y-1\right)\left(y+3\right)} have the same denominator, subtract them by subtracting their numerators.
\frac{2y-2-y^{2}-3y}{\left(y-1\right)\left(y+3\right)}+\frac{y^{2}+2}{y^{2}+2y-3}
Do the multiplications in 2\left(y-1\right)-y\left(y+3\right).
\frac{-y-2-y^{2}}{\left(y-1\right)\left(y+3\right)}+\frac{y^{2}+2}{y^{2}+2y-3}
Combine like terms in 2y-2-y^{2}-3y.
\frac{-y-2-y^{2}}{\left(y-1\right)\left(y+3\right)}+\frac{y^{2}+2}{\left(y-1\right)\left(y+3\right)}
Factor y^{2}+2y-3.
\frac{-y-2-y^{2}+y^{2}+2}{\left(y-1\right)\left(y+3\right)}
Since \frac{-y-2-y^{2}}{\left(y-1\right)\left(y+3\right)} and \frac{y^{2}+2}{\left(y-1\right)\left(y+3\right)} have the same denominator, add them by adding their numerators.
\frac{-y}{\left(y-1\right)\left(y+3\right)}
Combine like terms in -y-2-y^{2}+y^{2}+2.
\frac{-y}{y^{2}+2y-3}
Expand \left(y-1\right)\left(y+3\right).