急!急! F4 Polynomials

8.
Ax^2 + 3A + Bx + B + Cx^2 + 3Cx equivalent to 5x^2 + 10x + A
(A+C)x^2 + (B+3C)x + 3A + B equivalent to 5x^2 + 10x + A
comparing the coefficient at both side,
A+C = 5 => C = 5-A
B+3C = 10
3A+B = A => B = -2A
(-2A) + 3(5-A) = 10 => -5A + 15 = 10 => A = 1, B = -2, C = 4

9.
x^2 - 6x + 8 equivalent to x^2 - 8x + 15 - Ax + 3A - Bx + 4B
x^2 - 6x + 8 equivalent to x^2 - (8+A+B)x + (15 + 3A + 4B)
8+A+B = 6 => 3A+3B = -6
15+3A+4B = 8 => 3A+4B = -7
=> B = -1 and A = -1

10.
Ax^2 + (B+4)x + C equivalent to 2Bx^2 + 2(A-C)x + 1
A = 2B
B+4 = 2(A-C)
C = 1
B+4 = 2A-2 => B+6 = 2A
B+6 = 4B => B = 2 and A = 1

11a.
x^2 equivalent to A(x^2 - 3x + 2) + B(x^2 - 5x + 6) + C(x^2 - 4x + 3)
A+B+C = 1
-3A-5B-4C = 0
2A+6B +3C = 0 => A = 9/2, B = 1/2, C = -4
11b.
x^2 equivalent to (9/2)(x-2)(x-1) + (1/2)(x-2)-3) - 4(x-1)(x-3)
2x^2 equivalent to 9(x-2)(x-1) + (x-2)-3) - 8(x-1)(x-3)

12a.
2x^2 + 3x - 1 equivalent to A(x^2 + 3x + 2) + B(x^2 + 2x + 1) + C(x^2 + 4x + 4)
A+B+C = 2
3A+2B+4C = 3
2A+B+4C = -1 => A = 3, B = 1, C = -2
12b.
3(x+1)(x+2) - 4 equivalent to 2(x+2)^2 - (x+1)^2
3(x+1)(x+2)} - 2(x+2)^2 + (x+1)^2 equivalent to 4
from a, we have
2x^2 + 3x - 1 equivalent to 3(x+1)(x+2) + (x+1)^2 - 2(x+2)^2
therefore,
2x^2 + 3x - 1 = 4
2x^2 + 3x - 5 = 0
(2x + 5)(x - 1) = 0 => x = -5/2 or 1