Given a^2 + 3a - 1 = 0, the result for a^4 + 6a^3 + 12a^2 + 9a - 4 is required.

a^4 + 6a^3 + 12a^2 + 9a - 4

Let's split the terms and arrange them so that we can arrive at a^2 + 3a - 1 for every three terms by eliminating common factors.

=> a^4 + 3a^3 - a^2 + 3a^3 + 9a^2 - 3a + 3a^2 + 9a - 3 + a^2 + 3a - 1

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

=> 0

**The required value of a^4 + 6a^3 + 12a^2 + 9a - 4 = 0**