1. Problem: Factor each algebraic expression. 2. Formula and rules: Factoring involves finding common factors or patterns like difference of squares, grouping, or special products. 3. Part a) Expression: $2a^4b^5 - 5a^2b + 9ab$ - Identify common factors in all terms: each term has at least $a$ and $b$. - The smallest powers are $a^1$ and $b^1$, so factor out $ab$: $$2a^4b^5 - 5a^2b + 9ab = ab(2a^3b^4 - 5a + 9)$$ - No further factoring is possible inside the parentheses. 4. Part b) Expression: $-4m^2n + 12mn - 8m^5$ - Identify common factors: all terms have $m$. - The smallest power of $m$ is $m^1$, and $n$ is present in first two terms only. - Factor out $-4m$ (factoring out negative to simplify signs): $$-4m^2n + 12mn - 8m^5 = -4m(\cancel{m}n - \cancel{3}(-3)n + 2m^4)$$ - Correcting the cancellation step: $$-4m^2n + 12mn - 8m^5 = -4m(mn - 3n + 2m^4)$$ - No further factoring inside parentheses. Final answers: - a) $ab(2a^3b^4 - 5a + 9)$ - b) $-4m(mn - 3n + 2m^4)$