\( \begin{array}{ll}\text { (2) } & (x-1)\left(x^{2}-2 x+3\right) \\ \text { (4) } & (2 x-4)\left(x^{2}-3 x+1\right) \\ \text { (6) } & (a+2 b)\left(4 a^{2}-3 a b+b^{2}\right)\end{array} \)

Certainly! Let's expand each of the given expressions step by step. --- ### (2) \((x - 1)(x^{2} - 2x + 3)\) **Step-by-Step Expansion:** 1. **Distribute \(x\) over the second polynomial:** \[ x \cdot x^{2} = x^{3} \] \[ x \cdot (-2x) = -2x^{2} \] \[ x \cdot 3 = 3x \] 2. **Distribute \(-1\) over the second polynomial:** \[ -1 \cdot x^{2} = -x^{2} \] \[ -1 \cdot (-2x) = 2x \] \[ -1 \cdot 3 = -3 \] 3. **Combine all the terms:** \[ x^{3} - 2x^{2} + 3x - x^{2} + 2x - 3 \] 4. **Simplify by combining like terms:** \[ x^{3} - 3x^{2} + 5x - 3 \] **Expanded Form:** \[ x^{3} - 3x^{2} + 5x - 3 \] --- ### (4) \((2x - 4)(x^{2} - 3x + 1)\) **Step-by-Step Expansion:** 1. **Distribute \(2x\) over the second polynomial:** \[ 2x \cdot x^{2} = 2x^{3} \] \[ 2x \cdot (-3x) = -6x^{2} \] \[ 2x \cdot 1 = 2x \] 2. **Distribute \(-4\) over the second polynomial:** \[ -4 \cdot x^{2} = -4x^{2} \] \[ -4 \cdot (-3x) = 12x \] \[ -4 \cdot 1 = -4 \] 3. **Combine all the terms:** \[ 2x^{3} - 6x^{2} + 2x - 4x^{2} + 12x - 4 \] 4. **Simplify by combining like terms:** \[ 2x^{3} - 10x^{2} + 14x - 4 \] **Expanded Form:** \[ 2x^{3} - 10x^{2} + 14x - 4 \] --- ### (6) \((a + 2b)(4a^{2} - 3ab + b^{2})\) **Step-by-Step Expansion:** 1. **Distribute \(a\) over the second polynomial:** \[ a \cdot 4a^{2} = 4a^{3} \] \[ a \cdot (-3ab) = -3a^{2}b \] \[ a \cdot b^{2} = a b^{2} \] 2. **Distribute \(2b\) over the second polynomial:** \[ 2b \cdot 4a^{2} = 8a^{2}b \] \[ 2b \cdot (-3ab) = -6ab^{2} \] \[ 2b \cdot b^{2} = 2b^{3} \] 3. **Combine all the terms:** \[ 4a^{3} - 3a^{2}b + a b^{2} + 8a^{2}b - 6ab^{2} + 2b^{3} \] 4. **Simplify by combining like terms:** \[ 4a^{3} + ( -3a^{2}b + 8a^{2}b ) + ( a b^{2} - 6ab^{2} ) + 2b^{3} \] \[ 4a^{3} + 5a^{2}b - 5ab^{2} + 2b^{3} \] **Expanded Form:** \[ 4a^{3} + 5a^{2}b - 5ab^{2} + 2b^{3} \] --- If you have any specific questions or need further elaboration on any of these expansions, feel free to ask!