1. **State the problem:** Simplify the expression $$\frac{(a - x)^r x + bx + ax - b}{(x - ax + b)^r}$$. 2. **Rewrite the expression:** The numerator is $$(a - x)^r x + bx + ax - b$$ and the denominator is $$(x - ax + b)^r$$. 3. **Simplify the denominator:** Factor the denominator: $$x - ax + b = x(1 - a) + b$$ 4. **Simplify the numerator:** Group like terms: $$bx + ax = x(b + a)$$ So numerator becomes: $$(a - x)^r x + x(b + a) - b$$ 5. **Look for common factors or simplifications:** Note that $$(a - x) = -(x - a)$$, so: $$(a - x)^r = (-1)^r (x - a)^r$$ 6. **Rewrite numerator using this:** $$(-1)^r (x - a)^r x + x(b + a) - b$$ 7. **No further factorization is obvious without values of $a,b,r$.** **Final simplified form:** $$\frac{(-1)^r (x - a)^r x + x(b + a) - b}{(x(1 - a) + b)^r}$$ This is the simplified expression in terms of $a,b,r,x$.