Expand and simplify the following: \( \begin{array}{lll}\text { (a) }(x+4)^{2} & \text { (b) }(x-4)^{2} & \text { (c) }(2 x+4 y)^{2} \\ \text { (d) }(2 x-4 y)^{2} & \text { (c) }(-5 a-2 b)^{2} & \text { (f) }(a-3 b)^{3}\end{array} \)

Certainly! Let's expand and simplify each of the given expressions step by step. --- ### **a) \((x + 4)^2\)** **Expansion:** \[ (x + 4)^2 = (x)^2 + 2 \cdot x \cdot 4 + (4)^2 \] **Simplification:** \[ x^2 + 8x + 16 \] --- ### **b) \((x - 4)^2\)** **Expansion:** \[ (x - 4)^2 = (x)^2 - 2 \cdot x \cdot 4 + (4)^2 \] **Simplification:** \[ x^2 - 8x + 16 \] --- ### **c) \((2x + 4y)^2\)** **Expansion:** \[ (2x + 4y)^2 = (2x)^2 + 2 \cdot (2x) \cdot (4y) + (4y)^2 \] **Simplification:** \[ 4x^2 + 16xy + 16y^2 \] --- ### **d) \((2x - 4y)^2\)** **Expansion:** \[ (2x - 4y)^2 = (2x)^2 - 2 \cdot (2x) \cdot (4y) + (4y)^2 \] **Simplification:** \[ 4x^2 - 16xy + 16y^2 \] --- ### **e) \((-5a - 2b)^2\)** **Expansion:** \[ (-5a - 2b)^2 = (-5a)^2 + 2 \cdot (-5a) \cdot (-2b) + (-2b)^2 \] **Simplification:** \[ 25a^2 + 20ab + 4b^2 \] --- ### **f) \((a - 3b)^3\)** **Expansion Using the Binomial Theorem:** \[ (a - 3b)^3 = a^3 + 3a^2(-3b) + 3a(-3b)^2 + (-3b)^3 \] **Simplification:** \[ a^3 - 9a^2b + 27ab^2 - 27b^3 \] --- ### **Summary of Expanded and Simplified Expressions:** - **a)** \( (x + 4)^2 = x^2 + 8x + 16 \) - **b)** \( (x - 4)^2 = x^2 - 8x + 16 \) - **c)** \( (2x + 4y)^2 = 4x^2 + 16xy + 16y^2 \) - **d)** \( (2x - 4y)^2 = 4x^2 - 16xy + 16y^2 \) - **e)** \( (-5a - 2b)^2 = 25a^2 + 20ab + 4b^2 \) - **f)** \( (a - 3b)^3 = a^3 - 9a^2b + 27ab^2 - 27b^3 \) Feel free to ask if you need further explanations or additional assistance!