I am currently working on primes which can be expressed in form of a polynomial. For eg,
Find all primes which can be expressed in form $n^4-52n^2+595$
It is very essential to tell whether a polynomial can be factored or not. The polynomial above is very easy to factor but for some polynomials, it's extremely hard to tell.
Is there any way to prove whether a polynomial can be factored or not?
On
There are various advanced techniques to factor polynomials over the rationals or over the integers. See Wikipedia.
There is also a simple, well-known way to find if a polynomial of the form $$ f(x)=a_n x^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0 $$ and integer coefficients $a_0,\dots,a_n$ has linear factors over the rationals.
Indeed, if $x = p/q$ is a reduced solution of the equation $f(x)=0$, and equivalently $qx-p$ or $x-p/q$ is a factor of $f(x)$, then $p$ must be a (signed) divisor of $a_0$ and $q$ must be a divisor of $a_n$.
If $a_n=1$, then this means the only linear factors of $f(x)$ over the rationals, if they exist, are of the form $x+t$ where $t$ is a divisor of $a_0$.
For example, in your case the divisors of 595 are $$ \{{\pm1, \pm5, \pm7, \pm17, \pm35, \pm85, \pm119, \pm595}\} $$ thus if one of these values, say $t$, nullifies $t^4-52t^2+595$ then $x-t$ is a factor. If none of these values nullify $x^4-52x^2+595$, then you are sure there are no linear factors.
You can also check the Eisenstein's criterion.
As you mention, if the polynomial can be factored then it has only a finite number of prime values. Bunyakovski's conjecture is that otherwise, subject to some additional conditions, there are an infinite number of prime values. Although this is widely suspected to be true, there is no polynomial of degree greater than $1$ with integer coefficients that is known to give an infinite number of primes.