Explanation:
2^x < 2x³ + 1 ?
that would mean
x < log2(2x³ + 1)
x < log2(2×(x³ + 1/2))
x < log2(2) + log2(x³ + 1/2)
x < 1 + log2(x³ + 1/2)
x - 1 < log2(x³ + 1/2)
for large x the "-1" and the "+ 1/2" are becoming relatively irrelevant for the inequality.
and the limes of this goes to
x < log2(x³)
which is not true for large x, as the logarithm turns into a very flat curve, while "x" just keeps growing with slope 1 and therefore outpaces the logarithm.
for x = 20 for example, 2^x is already way larger than 2x³.
the "cross-over" point is between x=11 and x=12.
from there on 2^x will quickly become much larger than 2x³ + 1.