Muluply the following polynomials. \( \begin{array}{ll}\text { 5.) } 3 x\left(-2 x^{2}+3 x-5\right) & \text { 10.) }(-4 x+y)(4 x+y) \\ \text { 6.) }-4 x^{3}\left(x^{2}+2 x+8\right) & \text { 11.) }(2 x-7)^{2} \\ \text { 7.) }(4 x+6)(2 x-1) & \text { 12.) }(3 x+8)^{2} \\ \text { 8.) }(-x+y)(5 x-2 y) & \text { 13.) }(x+6)\left(4 x^{2}-2 x-1\right) \\ \text { 9.) }(5 x+1)(5 x-1) & \text { 14.) }\left(6 x^{2}-7 x-5\right)\left(x^{2}-5 x-8\right)\end{array} \)

Simplify the expression by following steps: - step0: Multiply the terms: \(\left(-4x+y\right)\left(4x+y\right)\) - step1: Reorder the terms: \(\left(y-4x\right)\left(y+4x\right)\) - step2: Simplify the product: \(y^{2}-\left(4x\right)^{2}\) - step3: Evaluate the power: \(y^{2}-16x^{2}\) Expand the expression \( -4 x^{3}(x^{2}+2 x+8) \) Simplify the expression by following steps: - step0: Calculate: \(-4x^{3}\left(x^{2}+2x+8\right)\) - step1: Rewrite the expression: \(-4\left(x^{2}+2x+8\right)x^{3}\) - step2: Rearrange the terms: \(\left(-4x^{2}-8x-32\right)x^{3}\) - step3: Apply the distributive property: \(-4x^{2}\times x^{3}-8x\times x^{3}-32x^{3}\) - step4: Multiply the terms: \(-4x^{5}-8x^{4}-32x^{3}\) Expand the expression \( (6 x^{2}-7 x-5)(x^{2}-5 x-8) \) Simplify the expression by following steps: - step0: Multiply the terms: \(\left(6x^{2}-7x-5\right)\left(x^{2}-5x-8\right)\) - step1: Apply the distributive property: \(6x^{2}\times x^{2}-6x^{2}\times 5x-6x^{2}\times 8-7x\times x^{2}-\left(-7x\times 5x\right)-\left(-7x\times 8\right)-5x^{2}-\left(-5\times 5x\right)-\left(-5\times 8\right)\) - step2: Multiply the terms: \(6x^{4}-30x^{3}-48x^{2}-7x^{3}-\left(-35x^{2}\right)-\left(-56x\right)-5x^{2}-\left(-25x\right)-\left(-40\right)\) - step3: Remove the parentheses: \(6x^{4}-30x^{3}-48x^{2}-7x^{3}+35x^{2}+56x-5x^{2}+25x+40\) - step4: Subtract the terms: \(6x^{4}-37x^{3}-18x^{2}+81x+40\) Expand the expression \( (4 x+6)(2 x-1) \) Simplify the expression by following steps: - step0: Multiply the terms: \(\left(4x+6\right)\left(2x-1\right)\) - step1: Apply the distributive property: \(4x\times 2x-4x\times 1+6\times 2x-6\times 1\) - step2: Multiply the terms: \(8x^{2}-4x+12x-6\) - step3: Add the terms: \(8x^{2}+8x-6\) Expand the expression \( (3 x+8)^{2 \) Simplify the expression by following steps: - step0: Calculate: \(\left(3x+8\right)^{2}\) - step1: Calculate: \(9x^{2}+48x+64\) Expand the expression \( (2 x-7)^{2 \) Simplify the expression by following steps: - step0: Calculate: \(\left(2x-7\right)^{2}\) - step1: Calculate: \(4x^{2}-28x+49\) Expand the expression \( 3 x(-2 x^{2}+3 x-5) \) Simplify the expression by following steps: - step0: Multiply the terms: \(3x\left(-2x^{2}+3x-5\right)\) - step1: Apply the distributive property: \(3x\left(-2x^{2}\right)+3x\times 3x-3x\times 5\) - step2: Multiply the terms: \(-6x^{3}+9x^{2}-15x\) Expand the expression \( (-x+y)(5 x-2 y) \) Simplify the expression by following steps: - step0: Multiply the terms: \(\left(-x+y\right)\left(5x-2y\right)\) - step1: Apply the distributive property: \(-x\times 5x-\left(-x\times 2y\right)+y\times 5x-y\times 2y\) - step2: Multiply the terms: \(-5x^{2}-\left(-2xy\right)+5yx-2y^{2}\) - step3: Remove the parentheses: \(-5x^{2}+2xy+5yx-2y^{2}\) - step4: Add the terms: \(-5x^{2}+7xy-2y^{2}\) Expand the expression \( (5 x+1)(5 x-1) \) Simplify the expression by following steps: - step0: Multiply the terms: \(\left(5x+1\right)\left(5x-1\right)\) - step1: Simplify the product: \(\left(5x\right)^{2}-1^{2}\) - step2: Evaluate the power: \(25x^{2}-1\) Expand the expression \( (x+6)(4 x^{2}-2 x-1) \) Simplify the expression by following steps: - step0: Multiply the terms: \(\left(x+6\right)\left(4x^{2}-2x-1\right)\) - step1: Apply the distributive property: \(x\times 4x^{2}-x\times 2x-x\times 1+6\times 4x^{2}-6\times 2x-6\times 1\) - step2: Multiply the terms: \(4x^{3}-2x^{2}-x+24x^{2}-12x-6\) - step3: Add the terms: \(4x^{3}+22x^{2}-13x-6\) Here are the results of multiplying the given polynomials: 1. **Problem 5:** \[ 3x(-2x^{2}+3x-5) = -6x^{3}+9x^{2}-15x \] 2. **Problem 6:** \[ -4x^{3}(x^{2}+2x+8) = -4x^{5}-8x^{4}-32x^{3} \] 3. **Problem 7:** \[ (4x+6)(2x-1) = 8x^{2}+8x-6 \] 4. **Problem 8:** \[ (-x+y)(5x-2y) = -5x^{2}+7xy-2y^{2} \] 5. **Problem 9:** \[ (5x+1)(5x-1) = 25x^{2}-1 \] 6. **Problem 10:** \[ (-4x+y)(4x+y) = y^{2}-16x^{2} \] 7. **Problem 11:** \[ (2x-7)^{2} = 4x^{2}-28x+49 \] 8. **Problem 12:** \[ (3x+8)^{2} = 9x^{2}+48x+64 \] 9. **Problem 13:** \[ (x+6)(4x^{2}-2x-1) = 4x^{3}+22x^{2}-13x-6 \] 10. **Problem 14:** \[ (6x^{2}-7x-5)(x^{2}-5x-8) = 6x^{4}-37x^{3}-18x^{2}+81x+40 \] These results represent the expanded forms of the products of the given polynomials.