Identities

1. x^3 - 6x^2 + 15x - 7 = (x+a)^3 + bx + c
(x + a)^3 + bx + c
= x^3 + 3ax^2 + 3a^2x + a^3 + bx + c
= x^3 + 3ax^2 + (3a^2 + b)x + (a^3 + c)
Comparing coeffients,
3a = -6 => a = -2
3a^2 + b = 15 => 12 + b = 15 => b = 3
a^3 + c = -7 => -8 + c = -7 => c = 1
2. a(x - 1)^2 + b(x + 1)^2 + c = x^2
a(x^2 - 2x + 1) + b(x^2 + 2x + 1) + c = x^2
(a + b)x^2 + (2b - 2a)x + (a + b + c ) = x^2
Comparing coefficients
a + b = 1 ... (1)
2b - 2a = 0 => a = b ... (2)
a + b + c = 0 ... (3)
Sub (2) into (1), 2a = 1 => a = 1/2 = b
(3) => 1/2 + 1/2 + c = 0 => c = -1

x^3-6x^2+15x-7=(x+a)^3+bx+c
a=(-6)/3=-2
x^3-6x^2+15x-7=(x-2)^3+bx+c
Set x=0
-7=-8+c
c=1
x^3-6x^2+15x-7=(x-2)^3+bx+1
Set x=1
1-6+15-7=-1+b+1
b=3