Hi Sveta.

You do this problem in two steps:

1. Find a and b.

f(x) + 2x + 8 = f(x+1)
ax² + bx + c + 2x + 8 = a(x+1)² + b(x+1) + c
ax² + (b+2)x + 8 + c = a(x² + 2x + 1) + bx + b + c
ax² + (b+2)x + 8 + c = ax² + 2ax + a + bx + b + c
ax² + (b+2)x + 8 + c = ax² + (2a+b)x + a + b + c
(b+2)x + 8 = (2a+b)x + a + b

So this can be separated into the x term and the units term:
b+2 = 2a+b
2a = 2
a = 1

and
8 = a + b

so substitute in a=1:
8 = 1 + b
b = 7

Now you have a and b.

2. Find c using f(1) = 12.

12 = f(1)
12 = a(1)² + b(1) + c
12= a + b + c
12 = 1 + 7 + c
c = 4

So f(x) = 1x² + 7x + 4

Now you have to check that no mistakes were made.
Verify that f(x) + 2x + 8 = f(x+1).

Stephen La Rocque.>