(2) \( \frac{a+b}{(a-2 b)^{2}}-\frac{1}{2 b-a} \)

To simplify the expression \( \frac{a+b}{(a-2b)^{2}} - \frac{1}{2b-a} \), we can find a common denominator, which is \((a-2b)^{2}(2b-a)\). First, let’s rewrite the fractions with the common denominator: 1. The first term becomes: \[ \frac{(a+b)(2b-a)}{(a-2b)^{2}(2b-a)} \] 2. The second term becomes: \[ \frac{(a-2b)^{2}}{(a-2b)^{2}(2b-a)} \] Now we combine the fractions over the common denominator: \[ \frac{(a+b)(2b-a) - (a-2b)^{2}}{(a-2b)^{2}(2b-a)} \] Next, we need to simplify the numerator: \[ (a+b)(2b-a) - (a-2b)^{2} \] Expanding the first term: \[ (a+b)(2b-a) = 2ab - a^2 + 2b^2 - ab = ab - a^2 + 2b^2 \] Now we expand the second term: \[ (a-2b)^{2} = a^2 - 4ab + 4b^2 \] Combining these, we have: \[ ab - a^2 + 2b^2 - (a^2 - 4ab + 4b^2) \] \[ = ab - a^2 + 2b^2 - a^2 + 4ab - 4b^2 \] \[ = (ab + 4ab) - 2a^2 + (2b^2 - 4b^2) \] \[ = 5ab - 2a^2 - 2b^2 \] Putting this back together, we get: \[ \frac{5ab - 2a^2 - 2b^2}{(a-2b)^{2}(2b-a)} \] And that’s the simplified expression!