1. a. Simplify \frac{m^2 - n^2}{n - m} Step 1: State the problem. Simplify the expression $ \frac{m^2 - n^2}{n - m} $. Step 2: Recognize the numerator as a difference of squares. $ m^2 - n^2 = (m - n)(m + n) $. Step 3: Rewrite the expression using this factorization: $$\frac{(m - n)(m + n)}{n - m}$$ Step 4: Notice that $ n - m = -(m - n) $, so: $$\frac{(m - n)(m + n)}{n - m} = \frac{(m - n)(m + n)}{-(m - n)}$$ Step 5: Cancel the common factor $ (m - n) $: $$\frac{\cancel{(m - n)}(m + n)}{-\cancel{(m - n)}} = - (m + n)$$ Step 6: Final simplified form: $$\boxed{-(m + n)}$$ b. Given that $ (2x^2 - x - 5)(x + 1) $ and $ 2x^3 + ax^2 - bx - 5 $ are identical polynomials, find $ a $ and $ b $. Step 1: Expand $ (2x^2 - x - 5)(x + 1) $: $$2x^2 \cdot x + 2x^2 \cdot 1 - x \cdot x - x \cdot 1 - 5 \cdot x - 5 \cdot 1 = 2x^3 + 2x^2 - x^2 - x - 5x - 5$$ Step 2: Simplify like terms: $$2x^3 + (2x^2 - x^2) + (-x - 5x) - 5 = 2x^3 + x^2 - 6x - 5$$ Step 3: Since the polynomials are identical: $$2x^3 + ax^2 - bx - 5 = 2x^3 + x^2 - 6x - 5$$ Step 4: Equate coefficients: $ a = 1 $ $ -b = -6 \Rightarrow b = 6 $ Step 5: Final answers: $$\boxed{a = 1, b = 6}$$