1. **State the problem:** Simplify the expression $$\frac{y^2 - 2ps + 2s}{2y - 4p + 4}$$. 2. **Identify the numerator and denominator:** - Numerator: $$y^2 - 2ps + 2s$$ - Denominator: $$2y - 4p + 4$$ 3. **Factor the denominator:** $$2y - 4p + 4 = 2(y - 2p + 2)$$ 4. **Try to factor the numerator:** Group terms: $$y^2 - 2ps + 2s = y^2 - 2ps + 2s$$ Notice the last two terms share a factor of $$2s$$: $$y^2 - 2ps + 2s = y^2 - 2ps + 2s$$ No obvious common factor for all terms, so try to factor by grouping: $$y^2 - 2ps + 2s = y^2 - 2ps + 2s$$ Rewrite as: $$y^2 - 2ps + 2s = y^2 - 2ps + 2s$$ No simpler factorization is apparent. 5. **Check if numerator can be factored as a product involving denominator's factor:** Denominator factor is $$y - 2p + 2$$. Try polynomial division or substitution. 6. **Perform polynomial division of numerator by $$y - 2p + 2$$:** Divide $$y^2 - 2ps + 2s$$ by $$y - 2p + 2$$ treating $$s$$ as a constant. 7. **Set up division:** Divide $$y^2$$ by $$y$$ to get $$y$$. Multiply divisor by $$y$$: $$y(y - 2p + 2) = y^2 - 2py + 2y$$ Subtract from numerator: $$ (y^2 - 2ps + 2s) - (y^2 - 2py + 2y) = -2ps + 2s + 2py - 2y$$ 8. **Simplify remainder:** Group terms: $$-2ps + 2py + 2s - 2y = 2p y - 2p s + 2s - 2y$$ Rewrite as: $$2p(y - s) + 2(s - y)$$ Note $$s - y = -(y - s)$$, so: $$2p(y - s) - 2(y - s) = (2p - 2)(y - s)$$ 9. **Divide remainder by divisor:** Since divisor is $$y - 2p + 2$$, and remainder is $$ (2p - 2)(y - s)$$, no direct cancellation unless $$y - s$$ equals divisor. 10. **Conclusion:** No exact factorization of numerator by denominator factor. 11. **Final simplified form:** Write denominator factored: $$\frac{y^2 - 2ps + 2s}{2(y - 2p + 2)}$$ No further simplification possible. **Answer:** $$\boxed{\frac{y^2 - 2ps + 2s}{2(y - 2p + 2)}}$$