By the Ratio Test, it converges for |x| < ½.

You consider the ratio | [(n+1)² 2^(n+1) x^(n+1)]/[n² 2^n x^n] | as n→∞ and determine the values of x for which this is less than 1.

The ratio simplifies to [((n+1)/n)²] * 2|x|

As n→∞ ((n+1)/n)² →1 so the series converges when 2|x| < 1, that is, for |x| < ½.