Infinite product and gamma functions

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How could one show or prove that

$\displaystyle \prod _{n=2}^{\infty } \frac{(2 n+x-2) (2 n+x-1)}{2 n (2 n+2 x-3)} =\frac{2^{x+1} \Gamma \left(x+\frac{1}{2}\right)}{\sqrt{\pi } \Gamma (x+2)} ?$