1. Factorize the odd-numbered problems: 1. Factorize $4x^2 - 2x^3 - 6x$: - Step 1: Rearrange terms as $-2x^3 + 4x^2 - 6x$. - Step 2: Factor out the greatest common factor (GCF) $-2x$: $$-2x(x^2 - 2x + 3)$$. - Step 3: The quadratic $x^2 - 2x + 3$ does not factor nicely, so the factorization is $$-2x(x^2 - 2x + 3)$$. 3. Factorize $12x^3y - 2x^2y^2 - 4xy$: - Step 1: Factor out the GCF $2xy$: $$2xy(6x^2 - xy - 2)$$. - Step 2: Inside the parentheses, try to factor $6x^2 - xy - 2$ by rearranging or grouping. - Step 3: Rewrite as $6x^2 - 3xy + 2xy - 2$ and factor by grouping: $$3x(2x - y) + 1(2x - y) = (3x +1)(2x - y)$$. - Step 4: So full factorization is $$2xy(3x + 1)(2x - y)$$. 5. Factorize $6x^3 - 9x^2 + 12x$: - Step 1: Factor out GCF 3x: $$3x(2x^2 - 3x + 4)$$. - Step 2: The quadratic does not factor easily, so factorization is $$3x(2x^2 - 3x + 4)$$. 7. Factorize $4x^2 - x^2 - 20x + 10$: - Step 1: Combine like terms: $3x^2 - 20x +10$. - Step 2: Find factors of $$3 \times 10 = 30$$ that sum to -20, which do not exist. - Step 3: Factor out GCF 1 (no change), expression is prime: $$3x^2 - 20x + 10$$. 9. Factorize $Ny+ yk - Lxk - xk$: - Step 1: Group terms: $(Ny + yk) - (Lxk + xk)$. - Step 2: Factor out $y$ from the first group and $xk$ from second: $$y(N + k) - xk(L + 1)$$. - Step 3: Cannot factor further without common terms. So final is $$y(N + k) - xk(L + 1)$$. 2. Expand and simplify the odd-numbered problems: 1. Expand $(6m - 1)(m + 5)$: - Step 1: Multiply terms: $$6m \times m = 6m^2$$, $$6m \times 5 = 30m$$, $$-1 \times m = -m$$, $$-1 \times 5 = -5$$. - Step 2: Combine: $$6m^2 + 30m - m - 5 = 6m^2 + 29m - 5$$. 3. Expand $2x^2 - [(2x-3)(3x+4)]$: - Step 1: First expand the bracket: $$(2x)(3x) = 6x^2$$, $$(2x)(4) = 8x$$, $$(-3)(3x) = -9x$$, $$(-3)(4) = -12$$. - Step 2: Sum terms inside bracket: $$6x^2 + 8x - 9x - 12 = 6x^2 - x - 12$$. - Step 3: Now simplify expression: $$2x^2 - (6x^2 - x - 12) = 2x^2 - 6x^2 + x + 12 = -4x^2 + x + 12$$. 5. Expand $(y + 3)^2$: - Step 1: This is a perfect square: $$(y + 3)^2 = y^2 + 2 \times y \times 3 + 3^2 = y^2 + 6y + 9$$. 3. Factorize the odd-numbered problems: 1. Factorize $4y^2 + 8y - 6y - 12$: - Step 1: Simplify middle terms: $4y^2 + 2y - 12$ - Step 2: Factor out GCF 2: $$2(2y^2 + y - 6)$$ - Step 3: Factor quadratic inside parentheses: Find two numbers that multiply to $2 \times -6 = -12$ and add to 1: 4 and -3. - Step 4: Rewrite as $2y^2 + 4y - 3y - 6$ and factor by grouping: $$2y(y + 2) - 3(y + 2) = (2y - 3)(y + 2)$$. - Step 5: Full factorization is $$2(2y - 3)(y + 2)$$. 5. Factorize $n^2 - nm + 3nm - 3m^2$: - Step 1: Combine like middle terms: $n^2 + 2nm - 3m^2$ - Step 2: Factor quadratic as: $$(n + 3m)(n - m)$$ (using ac method or grouping). Final answers included in factorizations and expansions above.