[Integer roots]

Let the positive integer roots be $p, q$. By Vieta, $p+q = k+3$ and $pq = 4k+1$. Substituting $k = p+q-3$:

$

pq = 4(p+q-3) + 1 = 4p + 4q - 11.

$

Rearrange to apply factor-shifting:

$

pq - 4p - 4q + 16 = 5 ;\Longrightarrow; (p-4)(q-4) = 5.

$

With $p, q \in \mathbb{Z}^{+}$, the factor pairs giving both $p, q \ge 1$ are $(p-4, q-4) \in \{(1,5),(5,1)\}$, yielding $(p,q) \in \{(5,9),(9,5)\}$. Both correspond to $k = 5+9-3 = 11$. Hence exactly one integer value of $k$ works. Answer: (B) 1.

Why CAT-level: Direct discriminant-perfect-square attempt is messy; the elegant route requires Simon-style factor shifting.