The problem is to expand and simplify the expression $(x+y-z)(x-y-z)$.

1. Problem Description

The problem is to expand and simplify the expression $(x+y-z)(x-y-z)$.

2. Solution Steps

We can rewrite the expression as follows:

$(x+y-z)(x-y-z) = (x-(z-y))(x-(y+z)) = ((x-z)+y)((x-z)-y)$.

Let $u = x-z$. Then the expression becomes

$(u+y)(u-y)$.

Using the difference of squares formula, $(a+b)(a-b) = a^2 - b^2$, we have

$(u+y)(u-y) = u^2 - y^2$.

Substituting $u = x-z$ back into the equation, we get

$(x-z)^2 - y^2 = (x^2 - 2xz + z^2) - y^2 = x^2 - 2xz + z^2 - y^2$.

Rearranging the terms, we get

$x^2 - y^2 + z^2 - 2xz$.

Alternative Approach:

We can expand the product term by term:

$(x+y-z)(x-y-z) = x(x-y-z) + y(x-y-z) - z(x-y-z)$

$= x^2 - xy - xz + yx - y^2 - yz - zx + zy + z^2$

$= x^2 - xy - xz + xy - y^2 - yz - xz + yz + z^2$

$= x^2 - y^2 + z^2 - 2xz$.

3. Final Answer

$x^2 - y^2 + z^2 - 2xz$