The given expression is (a/(a+1)) : (1 - 3a^2)/(1 - a^2)

To simplify this expression, we need to divide the numerator by the denominator separately:

a/(a+1) divided by (1 - 3a^2)/(1 - a^2)

To divide fractions, we multiply the first fraction by the reciprocal of the second fraction:

(a/(a+1)) * ((1 - a^2)/(1 - 3a^2))

Expanding the numerator and denominator:

= (a(1 - a^2))/((a + 1)(1 - 3a^2))

= (a - a^3)/(a - 3a^2 + a - 3a^3)

= (a - a^3)/(2a - 6a^2 - 3a^3)

We can further simplify by factoring out a common factor from the numerator and denominator:

= a(1 - a^2)/(a(2 - 6a - 3a^2))

= (1 - a^2)/(2 - 6a - 3a^2)

Therefore, the simplified expression is (1 - a^2)/(2 - 6a - 3a^2).