Which expression is equivalent to (x−2)(y+1)+(y−1)(x−2)?

To determine which expression is equivalent to ((x−2)(y+1)+(y−1)(x−2)), let's simplify the original expression step by step.

First, notice that both terms of the expression share a common factor of ((x−2)). We can factor this common term out:

[(x−2)(y+1) + (y−1)(x−2) = (x−2) \left( (y+1) + (y−1) \right).]

Next, simplify the expression inside the parentheses:

[(y+1) + (y−1) = y + 1 + y - 1 = 2y.]

Now substitute this back into the factored expression:

[(x−2)(2y).]

This simplifies to:

[2y(x−2).]

Thus, the expression ((x−2)(y+1)+(y−1)(x−2)) simplifies down to (2y(x−2)). Therefore, the correct expression that is equivalent to the original expression is indeed (2y(x−2)).