given a^2+3a-1=0, calculate a^4+6a^3+12a^2+9a-4?

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