The function $f(x)$ has the property that $f(x+y)=f(x)+f(y)+2 x y$, for all positive integers $x$ and $y$. If $f(1)=4$, then the numerical value of $f(8)$ is
(A) 72
(B) 84
(C) 88
(D) 64
(E) 80
Using the equation and information given,

$$
f(2)=f(1+1)=f(1)+f(1)+2(1)(1)=4+4+2=10
$$

(Notice that there is nothing else we could have actually calculated at this stage.)

Continuing,

$$
\begin{aligned}
& f(4)=f(2+2)=f(2)+f(2)+2(2)(2)=10+10+8=28 \\
& f(8)=f(4+4)=f(4)+f(4)+2(4)(4)=28+28+32=88
\end{aligned}
$$

Alternatively, we could have used the functional equation to calculate the values of $f(2)$, $f(3)=f(2+1), f(4)=f(3+1)$, etc., all the way up to $f(8)$.

ANSWER: (C)