A design engineer examines the quadratic expression: $P\left(x\right)=x^2+\left(k\right)x+41$, where $k$ is an integer. They want to solve for the integer linear factors, but they must note that $41$ is prime. For which integer value(s) of $k$ can $x^2+kx+41$ be factored into integers $(x+m)(x+n)$?

Any even integer $k$ will work because $41$ is odd so sums can be adjusted to even values for integer factors

$k=42$ or $k=-42$

$k=41$ only, because one of the factors must be $1$ and the other $41$ and their sum is $41$

No integer $k$ exists because $41$ is prime and prime constants never allow integer factorizations