1. **State the problem:** Simplify the given algebraic expressions by multiplying out the numerators and combining like terms. 2. **Recall the distributive property:** To multiply expressions like $(x-1)(x-1)$, use the formula $(a-b)^2 = a^2 - 2ab + b^2$. 3. **Simplify each numerator:** - For the first expression: $2x^2 - 3 + 3x^2 - 2 = (2x^2 + 3x^2) + (-3 - 2) = 5x^2 - 5$ 4. **Simplify the denominator:** - $(x-1)(x-1) = (x-1)^2 = x^2 - 2x + 1$ 5. **Write the simplified first expression:** $$\frac{5x^2 - 5}{(x-1)^2}$$ 6. **For the second expression numerator:** - $2x^2 - 3 + 3x^2 - 2 = 5x^2 - 5$ 7. **Denominator:** - $x(x-1)(x-1) = x(x-1)^2 = x(x^2 - 2x + 1) = x^3 - 2x^2 + x$ 8. **Simplified second expression:** $$\frac{5x^2 - 5}{x(x-1)^2}$$ 9. **For the third expression numerator:** - $2x^2 - x - 3 + 3x^2 - x - 2 = (2x^2 + 3x^2) + (-x - x) + (-3 - 2) = 5x^2 - 2x - 5$ 10. **Denominator:** - $x(x-1)(x-1) = x(x-1)^2$ 11. **Simplified third expression:** $$\frac{5x^2 - 2x - 5}{x(x-1)^2}$$ 12. **For the fourth expression numerator:** - $2x^2 - x + 3x^2 - 2 = (2x^2 + 3x^2) - x - 2 = 5x^2 - x - 2$ 13. **Denominator:** - $(x-1)(x-1) = (x-1)^2$ 14. **Simplified fourth expression:** $$\frac{5x^2 - x - 2}{(x-1)^2}$$ **Final answers:** 1. $$\frac{5x^2 - 5}{(x-1)^2}$$ 2. $$\frac{5x^2 - 5}{x(x-1)^2}$$ 3. $$\frac{5x^2 - 2x - 5}{x(x-1)^2}$$ 4. $$\frac{5x^2 - x - 2}{(x-1)^2}$$