To add or subtract expressions, expand them to make their denominators the same. Least common multiple of $\left(x-2\right)\left(x-1\right)$ and $\left(x-3\right)\left(x-2\right)$ is $\left(x-3\right)\left(x-2\right)\left(x-1\right)$. Multiply $\frac{x+3}{\left(x-2\right)\left(x-1\right)}$ times $\frac{x-3}{x-3}$. Multiply $\frac{x+1}{\left(x-3\right)\left(x-2\right)}$ times $\frac{x-1}{x-1}$.
Since $\frac{\left(x+3\right)\left(x-3\right)}{\left(x-3\right)\left(x-2\right)\left(x-1\right)}$ and $\frac{\left(x+1\right)\left(x-1\right)}{\left(x-3\right)\left(x-2\right)\left(x-1\right)}$ have the same denominator, add them by adding their numerators.
To add or subtract expressions, expand them to make their denominators the same. Least common multiple of $\left(x-2\right)\left(x-1\right)$ and $\left(x-3\right)\left(x-2\right)$ is $\left(x-3\right)\left(x-2\right)\left(x-1\right)$. Multiply $\frac{x+3}{\left(x-2\right)\left(x-1\right)}$ times $\frac{x-3}{x-3}$. Multiply $\frac{x+1}{\left(x-3\right)\left(x-2\right)}$ times $\frac{x-1}{x-1}$.
Since $\frac{\left(x+3\right)\left(x-3\right)}{\left(x-3\right)\left(x-2\right)\left(x-1\right)}$ and $\frac{\left(x+1\right)\left(x-1\right)}{\left(x-3\right)\left(x-2\right)\left(x-1\right)}$ have the same denominator, add them by adding their numerators.
