For how many integers $n$ between 1 and 100 does $x^2+x-n$ factor into the product of two linear factors with integer coefficients?
$\mathrm{(A) \ 0 } \qquad \mathrm{(B) \ 1 } \qquad \mathrm{(C) \ 2 } \qquad \mathrm{(D) \ 9 } \qquad \mathrm{(E) \ 10 }$
For $x^2+x-n$ to factor into a product of two linear factors, we must have $x^2+x-n = (x + a)(x + b)$, where $a$ and $b$ are integers.
By expansion of the product of the linear factors and comparison to the original quadratic,
$ab = -n$
$a + b = 1$.
The only way for this to work if $n$ is a positive integer is if $a = -b +1$.
Here are the possible pairs:


$a = -1, b = 2$
$a = -2, b = 3$
$\vdots$
$a = -9, b = 10$


This gives us 9 integers for $n$, $\boxed{\text{D}}$.