3y² + ky + 4

The only way to factor 3 is 3∙1 or 1∙3 but we can always reverse the
two binomial factors, so we might as well say any factorization is
like this:

(3y ± ?)(y ± ?)

The integers that go where the ?'s have the same sign and are 
pairs of factors that have product 4.  So they could be ±1 and ±4, ±2 and ±2, or
±4 and ±1

(3y ± 1)(y ± 4) = 3y² ± 7y + 4, so k = ±7

(3y ± 2)(y ± 2) = 3y² ± 8y + 4, so k = ±8

(3y ± 1)(y ± 4) = 3y² ± 13y + 4, so k = ±13

So there are 6 answers, 3 positive and 3 negative.

Edwin