\(=\dfrac{x+1}{x+3}+\dfrac{5}{x-3}-\dfrac{x^2+21}{\left(x-3\right)\left(x+3\right)}\)
\(=\dfrac{x^2-2x-3+5x+15-x^2-21}{\left(x-3\right)\left(x+3\right)}=\dfrac{3x-9}{\left(x-3\right)\left(x+3\right)}=\dfrac{3}{x+3}\)
\(\dfrac{x+1}{x+3}-\dfrac{5}{3-x}-\dfrac{x^2+21}{x^2-9}\)
\(=\dfrac{x+1}{x+3}+\dfrac{5}{x-3}-\dfrac{x^2+21}{\left(x-3\right)\left(x+3\right)}\)
\(=\dfrac{\left(x+1\right)\left(x-3\right)}{\left(x+3\right)\left(x-3\right)}+\dfrac{5\left(x+3\right)}{\left(x+3\right)\left(x-3\right)}-\dfrac{x^2+21}{\left(x+3\right)\left(x-3\right)}\)
\(=\dfrac{\left(x+1\right)\left(x-3\right)+5\left(x+3\right)-x^2-21}{\left(x-3\right)\left(x+3\right)}\)
\(=\dfrac{x^2-3x+x-3+5x+15-x^2-21}{\left(x-3\right)\left(x+3\right)}\)
\(=\dfrac{x^2-x^2-3x+x+5x-3+15-21}{\left(x-3\right)\left(x+3\right)}\)
\(=\dfrac{3x-9}{\left(x-3\right)\left(x+3\right)}\\ =\dfrac{3\left(x-3\right)}{\left(x-3\right)\left(x+3\right)}\\ =\dfrac{3}{x+3}\)
