smc

First simplify $f(x+2)$: $$f(x+2)=\frac{(x+2)(x+1)}{2}.$$ Then simplify the answers and see which one matches. $f(x)+f(2)=\frac{x(x-1)}{2}+1,$ $(x+2)f(x)=\frac{(x+2)x(x-1)}{2},$ $x(x+2)f(x)=\frac{(x+2)x^2(x-1)}{2},$ $\frac{xf(x)}{x+2}=\frac{x^2(x-1)}{2(x+2)},$ $\frac{(x+2)f(x+1)}{x}=\frac{(x+2)(x+1)}{2}.$ Since $f(x+2)=\frac{(x+2)f(x+1)}{x}=\frac{(x+2)(x+1)}{2}$, the answer is $\boxed{\frac{(x+2)f(x+1)}{x}}.$