I will do this by tomorrow’s 3 am if I can still remember and if no one else has done them yet.

a)

Like you wrote,

f(x) = (x - 4)(x + 1)(ax + b)


2f(1) = f(3) ①
f(-2) = -12 ②

For ①,
sub x = 1 for left side and x = 3 for right side

2(1 - 4)(1 + 1)(a(1) + b) = (3 - 4)(3 + 1)(3a + b)
-12a - 12b = -12a - 4b
-12b = -4b
8b = 0
b = 0


Sub b = 0 and x = -2 into ②,

(-2 - 4)(-2 + 1)(-2a + 0) = -12
-12a = -12
a = 1

So f(x) = x(x - 4)(x + 1)



b)

compare coefficient of term that is independent of x,

0 = -2 x 1 x 9 + C
0 = -18 + C
C = 18

compare coefficient of term in x⁴,
4 = 1 x 1 x B
B = 4

sub x = 1, C = 18, B = 4

4(1⁴) - 3(1³) + A(1) = (1 - 1)(1 + 2)(4(1²) - 4(1) + 9) + 18

1 + A = 18
(notice the factored part of the expression becomes zero as (1 - 1) = 0, so multiplying by 0 makes the factored part 0)

A = 17


So 4x⁴ - 3x² + 17x = (x - 1)(x + 2)(4x² - 4x + 9) + 18

Notice that (x - 1)(x + 2) = x² + x - 2


In the function,
The divisor can be taken as (x - 1)(x + 2), and the quotient as (4x² - 4x + 9). The remainder is 18.

Since dividing by (x² + x - 1) is the same as dividing by ((x - 1)(x + 2)),

the remainder when divided by (x² + x - 2) is just 18.