jmc

First, we try to simplify like terms. We find the prime factorizations of $45$ and $360$: $45=3^2\cdot 5$ and $360=2^3\cdot 3^2\cdot 5$. Hence, $$\sqrt{45}=\sqrt{3^2\cdot 5}=3\sqrt{5}$$and $$\begin{array}{rl}\sqrt{360}& =\sqrt{2^3\cdot 3^2\cdot 5}\\ & =\sqrt{(2\cdot 3{)}^2}\cdot \sqrt{2\cdot 5}=6\sqrt{2\cdot 5}.\end{array}$$Returning to the given expression, $$\begin{array}{rl}3\sqrt{5}-2\sqrt{5}+\frac{6\sqrt{2}\cdot \sqrt{5}}{\sqrt{2}}& =3\sqrt{5}-2\sqrt{5}+6\sqrt{5}\\ & =7\sqrt{5}=\sqrt{7^2\cdot 5}=\sqrt{245}.\end{array}$$Thus, $N=\boxed{245}$.