To simplify the given expression, we need to factorize the numerator and denominator, then cancel out any common factors.
The numerator can be factorized as the difference of two squares:
2 - 18m^2 = 2(1 - 9m^2) = 2(1 + 3m)(1 - 3m)
The denominator is already factorized:
1 + 3m
Now we can cancel out the common factor of (1 + 3m) from the numerator and denominator:
\[\frac{2-18m^2}{1+3m} = \frac{2(1+3m)(1-3m)}{1+3m} = 2(1-3m)\]
Therefore, the simplified form of the expression is 2(1-3m).