What is E(a) if a^2-3a+1=0 and E(x)=x^4-3x^3+4x^2-9x+3

We'll consider the expression E(x) and we'll re-write it
substituting 4x^2 by the sum 3x^2 + x^2

E(x) = x^4 - 3x^3 +
3x^2 + x^2 - 9x + 3

We'll group the terms in a convenient
way:

E(x) = (x^4 - 3x^3) + (3x^2 - 9x) + 3 +
x^2

We'll factorize:

E(x) =
x^3(x-3) + 3x(x-3) + x^2 + 3 (1)

Now, we'll consider the
given
constraint:

a^2-3a+1=0

We'll
factorize the first 2 terms:

a(a-3) + 1 =
0

We'll subtract 1 both
sides:

a(a-3) = -1 (2)

We also
could write

a^2-3a+1=0 as

a^2
= 3a - 1 (3)

We'll re-write (1) substituting x by
a:

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

We'll substitute (2) and (3) in
(1):

E(a) = a^2*(-1) + 3*(-1) + 3a - 1 +
3

E(a) = -3a + 1 - 3 + 3a - 1 +
3

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

E(a) =
0