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**

We'll re-write the constraint from enunciation:

a^2+3a-1=0 gives a^2+3a = 1

We'll raise to square both sides:

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

We'll expand the square:

a^4 + 6a^3 + 9a^2 = 1 (1)

We'll re-write the expresison that has to be calculated, with respect to (1):

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

We'll group the first 3 terms:

(a^4 + 6a^3 + 9a^2) + 3a^2 + 9a - 4

1 + 3a^2 + 9a - 4

We'll re-write -4 = -3-1

1 + 3a^2 + 9a - 3 - 1

We'll eliminate like terms and we'll get:

3a^2 + 9a - 3

We'll factorize by 3:

3(a^2 + 3a - 1)

But, from enunciation a^2 + 3a - 1 = 0, so the expresison to be calculated will be zero.

**a^4+6a^3+12a^2+9a-4 = 0**