The Diophantine equation 4X⁴-16X³+20X²-8X-3=Y²

Let’s solve the Diophantine equation: 4X⁴-16X³+20X²-8X-3=Y². Let’s suppose that X,Y are two integers satisfying the equation then: (2X²-4X-1)(2X²-4X+3)=Y². Let’s observe that 2X²-4X-1 and 2X²-4X+3 are coprime since if p is a possible common prime divisor it must be odd, let’s say it is p and we would have that -4=0 mod p therefore p=2, absurd. It follows that 2X²-4X-1=a² and 2X²-4X+3 = b² with a and b integers. Then a²-b²=-4 from which a²+4=b² which has no solutions in Z. It can be deduced that the initial Diophantine equation has no solutions.