1. Given expression: $2ax + 4bx + 8cx + 6dx$ Method: HCF Step 1: Identify the highest common factor (HCF) of all terms. Step 2: Coefficients: 2, 4, 8, 6. HCF is 2. Step 3: Each term contains $x$. Step 4: Factor out $2x$: $$2ax + 4bx + 8cx + 6dx = 2x(a + 2b + 4c + 3d)$$ 2. Given expression: $2ax + 4by + 3acx + 6bcy$ Method: Grouping Step 1: Group terms: $(2ax + 4by) + (3acx + 6bcy)$ Step 2: Factor each group: $$2ax + 4by = 2a x + 4b y$$ (no common factor between these two terms) Step 3: Factor second group: $$3acx + 6bcy = 3c (a x + 2 b y)$$ Step 4: Since no common binomial factor, try rearranging or factoring differently. Step 5: Group as $(2ax + 3acx) + (4by + 6bcy)$ Step 6: Factor each: $$x(2a + 3ac) + y(4b + 6bc) = x 2a(1 + rac{3c}{2}) + y 2b(2 + 3c)$$ Step 7: This is complicated; better to factor by grouping as: Group 1: $(2ax + 3acx) = x(2a + 3ac) = x a (2 + 3c)$ Group 2: $(4by + 6bcy) = 2b y (2 + 3c)$ Step 8: Now factor out common binomial $(2 + 3c)$: $$x a (2 + 3c) + 2b y (2 + 3c) = (2 + 3c)(a x + 2 b y)$$ 3. Given expression: $8yx - 4xy^2 - 6cw + 3cyw$ Method: Grouping Step 1: Group terms: $(8yx - 4xy^2) + (-6cw + 3cyw)$ Step 2: Factor each group: $$8yx - 4xy^2 = 4x y (2 - y)$$ $$-6cw + 3cyw = 3c w (-2 + y) = -3 c w (2 - y)$$ Step 3: Expression becomes: $$4 x y (2 - y) - 3 c w (2 - y)$$ Step 4: Factor out common binomial $(2 - y)$: $$(2 - y)(4 x y - 3 c w)$$ 4. Given expression: $8 y^2 x + 4 x^2 y^2 - 16 c x y^3 - 12 x c^2 y^4$ Method: Grouping Step 1: Group terms: $(8 y^2 x + 4 x^2 y^2) + (-16 c x y^3 - 12 x c^2 y^4)$ Step 2: Factor each group: $$8 y^2 x + 4 x^2 y^2 = 4 x y^2 (2 + x)$$ $$-16 c x y^3 - 12 x c^2 y^4 = -4 c x y^3 (4 + 3 c y)$$ Step 3: No common binomial factor; try rearranging. Step 4: Group as $(8 y^2 x - 16 c x y^3) + (4 x^2 y^2 - 12 x c^2 y^4)$ Step 5: Factor each: $$8 y^2 x - 16 c x y^3 = 8 x y^2 (1 - 2 c y)$$ $$4 x^2 y^2 - 12 x c^2 y^4 = 4 x y^2 (x - 3 c^2 y^2)$$ Step 6: No common binomial factor; factor out common $4 x y^2$: $$4 x y^2 (2 (1 - 2 c y) + x - 3 c^2 y^2)$$ Step 7: Simplify inside: $$2 - 4 c y + x - 3 c^2 y^2 = x + 2 - 4 c y - 3 c^2 y^2$$ Step 8: Final factorization: $$4 x y^2 (x + 2 - 4 c y - 3 c^2 y^2)$$ 5. Given expression: $18 a^4 y^3 x^4 - 54 a^5 y^4 x^5 + 72 a^3 x^3 y^3 - 36 a^3 y^3 x^6$ Method: HCF Step 1: Find HCF of coefficients: 18, 54, 72, 36 is 18. Step 2: Variables common to all terms: $a^3$, $y^3$, $x^3$ Step 3: Factor out $18 a^3 y^3 x^3$: $$18 a^3 y^3 x^3 (a x - 3 a^2 y x^2 + 4 - 2 x^3)$$ 6. Given expression: $3 p^3 m x^2 + 6 p^2 m^2 - 5 a p x^2 - 10 a m x$ Method: Grouping Step 1: Group terms: $(3 p^3 m x^2 + 6 p^2 m^2) + (-5 a p x^2 - 10 a m x)$ Step 2: Factor each group: $$3 p^3 m x^2 + 6 p^2 m^2 = 3 p^2 m (p x^2 + 2 m)$$ $$-5 a p x^2 - 10 a m x = -5 a (p x^2 + 2 m x)$$ Step 3: Notice $p x^2 + 2 m$ and $p x^2 + 2 m x$ differ; try factoring differently. Step 4: Group as $(3 p^3 m x^2 - 5 a p x^2) + (6 p^2 m^2 - 10 a m x)$ Step 5: Factor each: $$x^2 p (3 p^2 m - 5 a) + 2 m (3 p^2 m - 5 a x)$$ Step 6: No common binomial factor; try factoring by pairs: Step 7: Factor out $x$ from last term in second group: $$6 p^2 m^2 - 10 a m x = 2 m (3 p^2 m - 5 a x)$$ Step 8: No common binomial factor; final factorization is grouping: $$3 p^2 m (p x^2 + 2 m) - 5 a (p x^2 + 2 m x)$$ Final answers: 1. $2 x (a + 2 b + 4 c + 3 d)$ 2. $(2 + 3 c)(a x + 2 b y)$ 3. $(2 - y)(4 x y - 3 c w)$ 4. $4 x y^2 (x + 2 - 4 c y - 3 c^2 y^2)$ 5. $18 a^3 y^3 x^3 (a x - 3 a^2 y x^2 + 4 - 2 x^3)$ 6. $3 p^2 m (p x^2 + 2 m) - 5 a (p x^2 + 2 m x)$