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

Simplify the expression by following steps: - step0: Multiply the terms: \(\left(x+5\right)\left(x-2\right)\) - step1: Apply the distributive property: \(x\times x-x\times 2+5x-5\times 2\) - step2: Multiply the terms: \(x^{2}-2x+5x-10\) - step3: Add the terms: \(x^{2}+3x-10\) Expand the expression \( (x - 5)(x - 2) \) Simplify the expression by following steps: - step0: Multiply the terms: \(\left(x-5\right)\left(x-2\right)\) - step1: Apply the distributive property: \(x\times x-x\times 2-5x-\left(-5\times 2\right)\) - step2: Multiply the terms: \(x^{2}-2x-5x-\left(-10\right)\) - step3: Remove the parentheses: \(x^{2}-2x-5x+10\) - step4: Subtract the terms: \(x^{2}-7x+10\) Expand the expression \( (3 x - 1)(2 x + 3) \) Simplify the expression by following steps: - step0: Multiply the terms: \(\left(3x-1\right)\left(2x+3\right)\) - step1: Apply the distributive property: \(3x\times 2x+3x\times 3-2x-3\) - step2: Multiply the terms: \(6x^{2}+9x-2x-3\) - step3: Subtract the terms: \(6x^{2}+7x-3\) Expand the expression \( (2 x^{4} - 3 y^{2})(3 x^{4} + 2 y^{2}) \) Simplify the expression by following steps: - step0: Multiply the terms: \(\left(2x^{4}-3y^{2}\right)\left(3x^{4}+2y^{2}\right)\) - step1: Apply the distributive property: \(2x^{4}\times 3x^{4}+2x^{4}\times 2y^{2}-3y^{2}\times 3x^{4}-3y^{2}\times 2y^{2}\) - step2: Multiply the terms: \(6x^{8}+4x^{4}y^{2}-9y^{2}x^{4}-6y^{4}\) - step3: Subtract the terms: \(6x^{8}-5x^{4}y^{2}-6y^{4}\) Expand the expression \( (4 x^{4} + 3 y^{5})(2 x^{4} - 4 y^{3}) \) Simplify the expression by following steps: - step0: Multiply the terms: \(\left(4x^{4}+3y^{5}\right)\left(2x^{4}-4y^{3}\right)\) - step1: Apply the distributive property: \(4x^{4}\times 2x^{4}-4x^{4}\times 4y^{3}+3y^{5}\times 2x^{4}-3y^{5}\times 4y^{3}\) - step2: Multiply the terms: \(8x^{8}-16x^{4}y^{3}+6y^{5}x^{4}-12y^{8}\) Expand the expression \( 3 x (x + 3) - 3 a (3 a^{3} - 6 a^{2} + a) \) Simplify the expression by following steps: - step0: Subtract the terms: \(3x\left(x+3\right)-3a\left(3a^{3}-6a^{2}+a\right)\) - step1: Expand the expression: \(3x^{2}+9x-3a\left(3a^{3}-6a^{2}+a\right)\) - step2: Expand the expression: \(3x^{2}+9x-9a^{4}+18a^{3}-3a^{2}\) Expand the expression \( (7 m - 2)(x m)(3 m + 4 n) \) Simplify the expression by following steps: - step0: Calculate: \(\left(7m-2\right)\left(xm\right)\left(3m+4n\right)\) - step1: Multiply the terms: \(\left(7m-2\right)xm\left(3m+4n\right)\) - step2: Multiply the first two terms: \(xm\left(7m-2\right)\left(3m+4n\right)\) - step3: Multiply the terms: \(\left(7xm^{2}-2xm\right)\left(3m+4n\right)\) - step4: Apply the distributive property: \(7xm^{2}\times 3m+7xm^{2}\times 4n-2xm\times 3m-2xm\times 4n\) - step5: Multiply the terms: \(21xm^{3}+28xm^{2}n-6xm^{2}-8xmn\) Let's expand and simplify each expression step by step: 1. **Expression:** \( 3x(x+3) - 3a(3a^{3}-6a^{2}+a) \) - **Step 1:** Expand \( 3x(x+3) \): \[ 3x^2 + 9x \] - **Step 2:** Expand \( -3a(3a^{3}-6a^{2}+a) \): \[ -3a(3a^{3}) + 3a(6a^{2}) - 3a(a) = -9a^{4} + 18a^{3} - 3a^{2} \] - **Step 3:** Combine the results: \[ 3x^{2} + 9x - 9a^{4} + 18a^{3} - 3a^{2} \] **Final Result:** \( 3x^{2} + 9x - 9a^{4} + 18a^{3} - 3a^{2} \) 2. **Expression:** \( (x-5)(x-2) \) - **Step 1:** Expand: \[ x^2 - 2x - 5x + 10 = x^2 - 7x + 10 \] **Final Result:** \( x^{2} - 7x + 10 \) 3. **Expression:** \( (x+5)(x-2) \) - **Step 1:** Expand: \[ x^2 - 2x + 5x - 10 = x^2 + 3x - 10 \] **Final Result:** \( x^{2} + 3x - 10 \) 4. **Expression:** \( (3x-1)(2x+3) \) - **Step 1:** Expand: \[ 6x^2 + 9x - 2x - 3 = 6x^2 + 7x - 3 \] **Final Result:** \( 6x^{2} + 7x - 3 \) 5. **Expression:** \( (7m-2)(xm)(3m+4n) \) - **Step 1:** Expand: \[ 21xm^{3} + 28xm^{2}n - 6xm^{2} - 8xmn \] **Final Result:** \( 21xm^{3} + 28xm^{2}n - 6xm^{2} - 8xmn \) 6. **Expression:** \( (2x^{4}-3y^{2})(3x^{4}+2y^{2}) \) - **Step 1:** Expand: \[ 6x^{8} + 4x^{4}y^{2} - 9y^{4} \] **Final Result:** \( 6x^{8} - 5x^{4}y^{2} - 6y^{4} \) 7. **Expression:** \( (4x^{4}+3y^{5})(2x^{4}-4y^{3}) \) - **Step 1:** Expand: \[ 8x^{8} - 16x^{4}y^{3} + 6y^{5}x^{4} - 12y^{8} \] **Final Result:** \( 8x^{8} - 16x^{4}y^{3} + 6y^{5}x^{4} - 12y^{8} \) ### Summary of Results: 1. \( 3x^{2} + 9x - 9a^{4} + 18a^{3} - 3a^{2} \) 2. \( x^{2} - 7x + 10 \) 3. \( x^{2} + 3x - 10 \) 4. \( 6x^{2} + 7x - 3 \) 5. \( 21xm^{3} + 28xm^{2}n - 6xm^{2} - 8xmn \) 6. \( 6x^{8} - 5x^{4}y^{2} - 6y^{4} \) 7. \( 8x^{8} - 16x^{4}y^{3} + 6y^{5}x^{4} - 12y^{8} \)