Function f (x) = ax^{2}+bx+c, where a, b, and c are some constants. Define functions g and h as follows:

g (x) = f (x + 1) - f (x)

h (x) = g (x + 1) - g (x)

Find algebraic form of h (x)

Can anyone explain how to make it step by step?

G (x) = f (x+1) - f (x)

=[ a (x+1) ^2+b (x+1) + c ] - [ax^2+bx+c]

=[ a (x^2+2x+1) + bx + b + c ] - [ax^2 + bx + c]

=[ ax^2 + 2ax + a + bx + b + c ] - [ax^2 + bx + c]

= ax^2 + 2ax + a + bx + b + c - ax^2 - bx - c

= 2ax + a + b

Therefore g (x) = 2ax + a + b

h (x) = g (x+1) - g (x)

=2a (x+1) + a + b - [2ax+a+b]

=2ax + 1 + a + b - 2ax - a - b

Therefore h (x) = 1