Expand and simplify: \( \begin{array}{llllll}\text { (1) } \quad 3 x(x+3) & \text { (2) } & -3 a\left(3 a^{3}-6 a^{2}+a\right) & (3) & (x+5)(x+2) \\ \text { (4) } \quad(x-5)(x-2) & (5) & (x+5)(x-2) & (4) & (x-5)(x+2) \\ \text { (7) } & (3 x-1)(2 x+3) & \text { (8) } & (7 m-2 n)(3 m+4 n) & \\ \text { (9) }\left(2 x^{4}-3 y^{2}\right)\left(3 x^{4}+2 y^{2}\right) & \text { (10) }\left(4 x^{4}+3 y^{5}\right)\left(2 x^{4}-4 y^{3}\right)\end{array} \)

Certainly! Let's expand and simplify each of the given expressions step by step. --- ### **(1) \( 3x(x + 3) \)** **Expansion:** \[ 3x \cdot x + 3x \cdot 3 \] **Simplification:** \[ 3x^2 + 9x \] **Final Answer:** \[ 3x^2 + 9x \] --- ### **(2) \( -3a(3a^{3} - 6a^{2} + a) \)** **Expansion:** \[ -3a \cdot 3a^3 + (-3a) \cdot (-6a^2) + (-3a) \cdot a \] **Simplification:** \[ -9a^4 + 18a^3 - 3a^2 \] **Final Answer:** \[ -9a^4 + 18a^3 - 3a^2 \] --- ### **(3) \( (x + 5)(x + 2) \)** **Expansion (Using FOIL Method):** \[ x \cdot x + x \cdot 2 + 5 \cdot x + 5 \cdot 2 \] **Simplification:** \[ x^2 + 2x + 5x + 10 = x^2 + 7x + 10 \] **Final Answer:** \[ x^2 + 7x + 10 \] --- ### **(4) \( (x - 5)(x - 2) \)** **Expansion (Using FOIL Method):** \[ x \cdot x + x \cdot (-2) + (-5) \cdot x + (-5) \cdot (-2) \] **Simplification:** \[ x^2 - 2x - 5x + 10 = x^2 - 7x + 10 \] **Final Answer:** \[ x^2 - 7x + 10 \] --- ### **(5) \( (x + 5)(x - 2) \)** **Expansion (Using FOIL Method):** \[ x \cdot x + x \cdot (-2) + 5 \cdot x + 5 \cdot (-2) \] **Simplification:** \[ x^2 - 2x + 5x - 10 = x^2 + 3x - 10 \] **Final Answer:** \[ x^2 + 3x - 10 \] --- ### **(6) \( (x - 5)(x + 2) \)** **Expansion (Using FOIL Method):** \[ x \cdot x + x \cdot 2 + (-5) \cdot x + (-5) \cdot 2 \] **Simplification:** \[ x^2 + 2x - 5x - 10 = x^2 - 3x - 10 \] **Final Answer:** \[ x^2 - 3x - 10 \] --- ### **(7) \( (3x - 1)(2x + 3) \)** **Expansion:** \[ 3x \cdot 2x + 3x \cdot 3 + (-1) \cdot 2x + (-1) \cdot 3 \] **Simplification:** \[ 6x^2 + 9x - 2x - 3 = 6x^2 + 7x - 3 \] **Final Answer:** \[ 6x^2 + 7x - 3 \] --- ### **(8) \( (7m - 2n)(3m + 4n) \)** **Expansion:** \[ 7m \cdot 3m + 7m \cdot 4n + (-2n) \cdot 3m + (-2n) \cdot 4n \] **Simplification:** \[ 21m^2 + 28mn - 6mn - 8n^2 = 21m^2 + 22mn - 8n^2 \] **Final Answer:** \[ 21m^2 + 22mn - 8n^2 \] --- ### **(9) \( (2x^{4} - 3y^{2})(3x^{4} + 2y^{2}) \)** **Expansion:** \[ 2x^4 \cdot 3x^4 + 2x^4 \cdot 2y^2 + (-3y^2) \cdot 3x^4 + (-3y^2) \cdot 2y^2 \] **Simplification:** \[ 6x^8 + 4x^4y^2 - 9x^4y^2 - 6y^4 = 6x^8 - 5x^4y^2 - 6y^4 \] **Final Answer:** \[ 6x^8 - 5x^4y^2 - 6y^4 \] --- ### **(10) \( (4x^{4} + 3y^{5})(2x^{4} - 4y^{3}) \)** **Expansion:** \[ 4x^4 \cdot 2x^4 + 4x^4 \cdot (-4y^3) + 3y^5 \cdot 2x^4 + 3y^5 \cdot (-4y^3) \] **Simplification:** \[ 8x^8 - 16x^4y^3 + 6x^4y^5 - 12y^8 \] **Final Answer:** \[ 8x^8 + 6x^4y^5 - 16x^4y^3 - 12y^8 \] --- These expanded and simplified forms should help clarify each expression!