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\frac{x}{x+3}+\frac{8x+18}{\left(x+1\right)\left(x+3\right)}
Factor x^{2}+4x+3.
\frac{x\left(x+1\right)}{\left(x+1\right)\left(x+3\right)}+\frac{8x+18}{\left(x+1\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+1\right)\left(x+3\right) is \left(x+1\right)\left(x+3\right). Multiply \frac{x}{x+3} times \frac{x+1}{x+1}.
\frac{x\left(x+1\right)+8x+18}{\left(x+1\right)\left(x+3\right)}
Since \frac{x\left(x+1\right)}{\left(x+1\right)\left(x+3\right)} and \frac{8x+18}{\left(x+1\right)\left(x+3\right)} have the same denominator, add them by adding their numerators.
\frac{x^{2}+x+8x+18}{\left(x+1\right)\left(x+3\right)}
Do the multiplications in x\left(x+1\right)+8x+18.
\frac{x^{2}+9x+18}{\left(x+1\right)\left(x+3\right)}
Combine like terms in x^{2}+x+8x+18.
\frac{\left(x+3\right)\left(x+6\right)}{\left(x+1\right)\left(x+3\right)}
Factor the expressions that are not already factored in \frac{x^{2}+9x+18}{\left(x+1\right)\left(x+3\right)}.
\frac{x+6}{x+1}
Cancel out x+3 in both numerator and denominator.
\frac{x}{x+3}+\frac{8x+18}{\left(x+1\right)\left(x+3\right)}
Factor x^{2}+4x+3.
\frac{x\left(x+1\right)}{\left(x+1\right)\left(x+3\right)}+\frac{8x+18}{\left(x+1\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+1\right)\left(x+3\right) is \left(x+1\right)\left(x+3\right). Multiply \frac{x}{x+3} times \frac{x+1}{x+1}.
\frac{x\left(x+1\right)+8x+18}{\left(x+1\right)\left(x+3\right)}
Since \frac{x\left(x+1\right)}{\left(x+1\right)\left(x+3\right)} and \frac{8x+18}{\left(x+1\right)\left(x+3\right)} have the same denominator, add them by adding their numerators.
\frac{x^{2}+x+8x+18}{\left(x+1\right)\left(x+3\right)}
Do the multiplications in x\left(x+1\right)+8x+18.
\frac{x^{2}+9x+18}{\left(x+1\right)\left(x+3\right)}
Combine like terms in x^{2}+x+8x+18.
\frac{\left(x+3\right)\left(x+6\right)}{\left(x+1\right)\left(x+3\right)}
Factor the expressions that are not already factored in \frac{x^{2}+9x+18}{\left(x+1\right)\left(x+3\right)}.
\frac{x+6}{x+1}
Cancel out x+3 in both numerator and denominator.