To solve the problem, we need to find the function (g(x)) that is symmetric to (f(x)) about the point ((2,0)), then compute (a+b+c) (which equals (g(1))).

Key Insight: Symmetry About a Point

If ((x,y)) is on (g(x)), its symmetric point ((x',y')) about ((2,0)) lies on (f(x)). Using midpoint formula:

• (x' = 4 - x) (midpoint of (x) and (x') is 2)
• (y' = -y) (midpoint of (y) and (y') is 0)

Since ((x',y')\in f(x)):
(y' = f(x') \implies -y = f(4-x) \implies g(x) = -f(4-x)).

Compute (a+b+c = g(1))

(g(1) = -f(4-1) = -f(3)).

Calculate (f(3)):
(f(3) = 3^2 -2\cdot3 +2 = 9-6+2=5).

Thus:
(g(1) = -5).

Answer: (\boxed{-5})

作者声明:本文包含人工智能生成内容。