To solve this expression, let's simplify step by step:

[tex]\left(\frac{2-a}{1-a}\right)^2 - a \times \frac{a^2 - 2a + 1}{a - 2}[/tex]

First, simplify the term inside the parentheses:

[tex]\left(\frac{2-a}{1-a}\right)^2 = \left(\frac{2-a}{1-a}\right) \times \left(\frac{2-a}{1-a}\right) = \frac{(2-a)^2}{(1-a)^2}[/tex]

Now expand ((2-a)^2) and ((1-a)^2):

[tex]\frac{(2-a)^2}{(1-a)^2} = \frac{4 - 4a + a^2}{1 - 2a + a^2} = \frac{a^2 - 4a + 4}{a^2 - 2a + 1} = \frac{(a - 2)^2}{(a - 1)^2}[/tex]

Substitute this back into the original expression:

[tex]\frac{(a - 2)^2}{(a - 1)^2} - a \times \frac{a^2 - 2a + 1}{a - 2}[/tex]

Now expand the expression:

[tex]\frac{(a - 2)^2}{(a - 1)^2} = \frac{a^2 - 4a + 4}{a^2 - 2a + 1} = \frac{a^2 - 4a + 4}{(a - 1)^2}[/tex]

Then simplify the entire expression:

[tex]\frac{a^2 - 4a + 4}{(a - 1)^2} - a \times \frac{a^2 - 2a + 1}{a - 2} = \frac{a^2 - 4a + 4}{(a - 1)^2} - \frac{a^3 - 2a^2 + a}{a - 2}[/tex]

I hope this helps! Let me know if you need any further assistance.