how to Find factorizations of 974!+2 by using algorithms
I suspect this number has a lot of possible factorizations. You could use the elliptic curve factorization algorithm, it works remarkably smoothly for numbers with many possible factorizations.
Or maybe there's some arithmetic trick to simplify this, I do not know.
It should be obvious that 2 is a factor.
$\displaystyle 2 \left( \dfrac{974!}{2} + 1\right) = 974! + 2$
Test $\displaystyle \left( \dfrac{974!}{2} + 1\right)$ for primality.
Most likely it is a prime number.
I did a Fermat test for several hundred primes.
ALL results indicated that the number is prime.