F.2 identity

Ax^2-Ax+Bx^2-B+Cx^2+Cx=x^2+4
(A+B+C)x^2+(-A+C)x-B=x^2+4

so,
A+B+C = 1 --- eq. 1

-A+C = 0 --> A=C

-B=4 -->B=-4

therefore, eq.1 --> 2A = 1+4 --> A = 5/2 = C

A=5/2 , B= -4, C= 5/2

Ax(x-1)+B(x+1)(x-1)+C(x+1)x=x^2+4

Ax^2-A+Bx^2-B+Cx+C=x^2+4

(A+B)x^2+(C-A)x+(C-B) = x^2+4

Therefore, by Observation,

Eq 1: A+B =1
Eq 2: C-A=0
Eq 3: C-B=4

Eq 2 - Eq 3 , we have A-B=4 , B = A-4

Sub B =A-4 into Eq.1

A+A-4 =1
2A=5
A=2.5

Therefore, B = -1.5
Therefore, C = 2.5

Ax(x-1)+B(x+1)(x-1)+C(x+1)x=x^2+4
=> A(x^2 - x) + B(x^2 - 1) + C(x^2 + x) = x^2 + 4
=> (A+B+C)x^2 + (C-A)x - B = x^2 + 0 x + 4
=> A+B+C = 1 -------------(1)
C-A = 0 => A = C ------------(2)
-B = 4 or B = -4 -------------(3)

By substituting (2) and (3) into 1, we have
C+(-4)+C = 1
=> 2C = 5
=> C = 5/2
By (2), we have A = 5/2

In conclusion, A = 5/2, B = -4 and C = 5/2

Ax(x-1)+B(x+1)(x-1)+C(x+1)x=x^2+4

Ax^2-A+Bx^2-B+Cx+C=x^2+4

(A+B)x^2+(C-A)x+(C-B) = x^2+4

Therefore, by Observation,

Eq 1: A+B =1
Eq 2: C-A=0
Eq 3: C-B=4

Eq 2 - Eq 3 , we have A-B=4 , B = A-4

Sub B =A-4 into Eq.1

A+A-4 =1
2A=5
A=2.5

Therefore, B = -1.5
Therefore, C = 2.5

Ax^2-Ax+Bx^2-B+Cx^2+Cx=x^2+4
(A+B+C)x^2+(-A+C)x-B=x^2+4

so,
A+B+C = 1 --- eq. 1

-A+C = 0 --> A=C

-B=4 -->B=-4

therefore, eq.1 --> 2A = 1+4 --> A = 5/2 = C

A=5/2 , B= -4, C= 5/2