1. **Problem statement:** Factor the expression $$ab + 2ac + 3b^2 + 6bc - 5a - 13b + 4c - 10$$ 2. **Exercise (1) with variable $a$:** Arrange the expression in standard polynomial form with $a$ as the variable: $$ab + 2ac + 3b^2 + 6bc - 5a - 13b + 4c - 10 = b(a + 6c - 13) + (2ac - 5a + 4c - 10)$$ Rewrite grouping terms with $a$: $$= a(b + 2c - 5) + (3b^2 + 6bc - 13b + 4c - 10)$$ 3. **Factor the constant part (without $a$):** Group terms: $$3b^2 + 6bc - 13b + 4c - 10 = 3b^2 + 6bc - 13b + 4c - 10$$ Try grouping: $$= 3b^2 + 6bc - 13b + 4c - 10$$ Group as: $$(3b^2 + 6bc) + (-13b + 4c - 10)$$ Factor $3b$ from first group: $$3b(b + 2c) + (-13b + 4c - 10)$$ No obvious common factor in second group, so try factoring by grouping differently or consider the entire expression. 4. **Exercise (2) with variable $b$:** Arrange the expression in standard polynomial form with $b$ as the variable: $$ab + 2ac + 3b^2 + 6bc - 5a - 13b + 4c - 10$$ Group terms with $b$: $$= 3b^2 + b(a + 6c - 13) + (2ac - 5a + 4c - 10)$$ 5. **Factor as a quadratic in $b$:** The quadratic in $b$ is: $$3b^2 + (a + 6c - 13)b + (2ac - 5a + 4c - 10)$$ 6. **Use quadratic formula to factor if possible:** Discriminant: $$\Delta = (a + 6c - 13)^2 - 4 \times 3 \times (2ac - 5a + 4c - 10)$$ Simplify: $$= (a + 6c - 13)^2 - 12(2ac - 5a + 4c - 10)$$ 7. **Summary:** - For (1), the expression arranged with $a$ is: $$a(b + 2c - 5) + (3b^2 + 6bc - 13b + 4c - 10)$$ - For (2), the expression arranged with $b$ is: $$3b^2 + b(a + 6c - 13) + (2ac - 5a + 4c - 10)$$ Further factorization depends on values of $a,c$ or specific factorization techniques. **Final answers:** (1) $$a(b + 2c - 5) + 3b^2 + 6bc - 13b + 4c - 10$$ (2) $$3b^2 + b(a + 6c - 13) + 2ac - 5a + 4c - 10$$