The problem asks us to expand the expression $(a - b + c)^2$.

1. Problem Description

The problem asks us to expand the expression $(a - b + c)^2$.

2. Solution Steps

We can expand the expression $(a - b + c)^2$ by multiplying it by itself:

$(a - b + c)^2 = (a - b + c)(a - b + c)$

Now, we distribute each term in the first parentheses to each term in the second parentheses:

$= a(a - b + c) - b(a - b + c) + c(a - b + c)$

$= a^2 - ab + ac - ba + b^2 - bc + ca - cb + c^2$

Combine like terms:

$= a^2 + b^2 + c^2 - ab - ba + ac + ca - bc - cb$

Since $ab = ba$, $ac = ca$, and $bc = cb$, we can simplify further:

$= a^2 + b^2 + c^2 - 2ab + 2ac - 2bc$

Alternatively, we can use the formula $(x + y + z)^2 = x^2 + y^2 + z^2 + 2xy + 2xz + 2yz$.

In our case, $x = a$, $y = -b$, and $z = c$.

So, $(a - b + c)^2 = a^2 + (-b)^2 + c^2 + 2(a)(-b) + 2(a)(c) + 2(-b)(c)$

$= a^2 + b^2 + c^2 - 2ab + 2ac - 2bc$

3. Final Answer

The expansion of $(a - b + c)^2$ is $a^2 + b^2 + c^2 - 2ab + 2ac - 2bc$.