\(\left(x^2-y^2+6x+9\right):\left(x+y+3\right)\)
\(=\text{[}\left(x^2+6x+9\right)-y^2\text{]}:\left(x+y+3\right)\)
\(=\text{[}\left(x+3\right)^2-y^2\text{]}:\left(x+y+3\right)\)
\(=\left(x+3+y\right)\left(x+3-y\right):\left(x+y+3\right)\)
\(=x+3-y\)