\( \begin{array}{llll}\text { 11. } & 12 x^{2}+13 x+3 & \text { 12. } & 21 x^{2}-x-2 \\ \text { 13. } & 12 x^{2}+5 x-3 & \text { 14. } & 6 x^{2}-23 x+7 \\ \text { 15. } & 2 x^{2}+5 x-12 & \text { 16. } & 3 x^{2}+11 x-20 \\ \text { 17. } 7 x^{2}+5 x-2 & \text { 18. } & 5 x^{2}+17 x+6 \\ \text { 19. } 13 x^{2}-27 x+2 & \text { 20. } & 2 x^{2}-7 x+6 \\ \text { 21. } 6 x^{2}+7 x-5 & \text { 22. } & 8 x^{2}-14 x+3\end{array} \)

Sure! It looks like you have a list of quadratic expressions from problems 11 to 22. Below, each quadratic is factored into the product of two binomials. --- ### 11. \(12x^{2} + 13x + 3\) **Factored Form:** \((3x + 1)(4x + 3)\) **Steps:** 1. Multiply the coefficient of \(x^2\) and the constant term: \(12 \times 3 = 36\). 2. Find two numbers that multiply to 36 and add to 13. These numbers are 9 and 4. 3. Rewrite the middle term using these numbers: \(12x^2 + 9x + 4x + 3\). 4. Factor by grouping: \[ 3x(4x + 3) + 1(4x + 3) = (3x + 1)(4x + 3) \] --- ### 12. \(21x^{2} - x - 2\) **Factored Form:** \((7x + 2)(3x - 1)\) **Steps:** 1. Multiply the coefficient of \(x^2\) and the constant term: \(21 \times (-2) = -42\). 2. Find two numbers that multiply to -42 and add to -1. These numbers are -7 and 6. 3. Rewrite the middle term: \(21x^2 - 7x + 6x - 2\). 4. Factor by grouping: \[ 7x(3x - 1) + 2(3x - 1) = (7x + 2)(3x - 1) \] --- ### 13. \(12x^{2} + 5x - 3\) **Factored Form:** \((3x - 1)(4x + 3)\) **Steps:** 1. Multiply the coefficient of \(x^2\) and the constant term: \(12 \times (-3) = -36\). 2. Find two numbers that multiply to -36 and add to 5. These numbers are 9 and -4. 3. Rewrite the middle term: \(12x^2 + 9x - 4x - 3\). 4. Factor by grouping: \[ 3x(4x + 3) - 1(4x + 3) = (3x - 1)(4x + 3) \] --- ### 14. \(6x^{2} - 23x + 7\) **Factored Form:** \((3x - 1)(2x - 7)\) **Steps:** 1. Multiply the coefficient of \(x^2\) and the constant term: \(6 \times 7 = 42\). 2. Find two numbers that multiply to 42 and add to -23. These numbers are -21 and -2. 3. Rewrite the middle term: \(6x^2 - 21x - 2x + 7\). 4. Factor by grouping: \[ 3x(2x - 7) - 1(2x - 7) = (3x - 1)(2x - 7) \] --- ### 15. \(2x^{2} + 5x - 12\) **Factored Form:** \((2x - 3)(x + 4)\) **Steps:** 1. Multiply the coefficient of \(x^2\) and the constant term: \(2 \times (-12) = -24\). 2. Find two numbers that multiply to -24 and add to 5. These numbers are 8 and -3. 3. Rewrite the middle term: \(2x^2 + 8x - 3x - 12\). 4. Factor by grouping: \[ 2x(x + 4) - 3(x + 4) = (2x - 3)(x + 4) \] --- ### 16. \(3x^{2} + 11x - 20\) **Factored Form:** \((3x - 4)(x + 5)\) **Steps:** 1. Multiply the coefficient of \(x^2\) and the constant term: \(3 \times (-20) = -60\). 2. Find two numbers that multiply to -60 and add to 11. These numbers are 15 and -4. 3. Rewrite the middle term: \(3x^2 + 15x - 4x - 20\). 4. Factor by grouping: \[ 3x(x + 5) - 4(x + 5) = (3x - 4)(x + 5) \] --- ### 17. \(7x^{2} + 5x - 2\) **Factored Form:** \((7x - 2)(x + 1)\) **Steps:** 1. Multiply the coefficient of \(x^2\) and the constant term: \(7 \times (-2) = -14\). 2. Find two numbers that multiply to -14 and add to 5. These numbers are 7 and -2. 3. Rewrite the middle term: \(7x^2 + 7x - 2x - 2\). 4. Factor by grouping: \[ 7x(x + 1) - 2(x + 1) = (7x - 2)(x + 1) \] --- ### 18. \(5x^{2} + 17x + 6\) **Factored Form:** \((5x + 2)(x + 3)\) **Steps:** 1. Multiply the coefficient of \(x^2\) and the constant term: \(5 \times 6 = 30\). 2. Find two numbers that multiply to 30 and add to 17. These numbers are 15 and 2. 3. Rewrite the middle term: \(5x^2 + 15x + 2x + 6\). 4. Factor by grouping: \[ 5x(x + 3) + 2(x + 3) = (5x + 2)(x + 3) \] --- ### 19. \(13x^{2} - 27x + 2\) **Factored Form:** \((13x - 1)(x - 2)\) **Steps:** 1. Multiply the coefficient of \(x^2\) and the constant term: \(13 \times 2 = 26\). 2. Find two numbers that multiply to 26 and add to -27. These numbers are -26 and -1. 3. Rewrite the middle term: \(13x^2 - 26x - x + 2\). 4. Factor by grouping: \[ 13x(x - 2) - 1(x - 2) = (13x - 1)(x - 2) \] --- ### 20. \(2x^{2} - 7x + 6\) **Factored Form:** \((x - 2)(2x - 3)\) **Steps:** 1. Multiply the coefficient of \(x^2\) and the constant term: \(2 \times 6 = 12\). 2. Find two numbers that multiply to 12 and add to -7. These numbers are -3 and -4. 3. Rewrite the middle term: \(2x^2 - 3x - 4x + 6\). 4. Factor by grouping: \[ x(2x - 3) - 2(2x - 3) = (x - 2)(2x - 3) \] --- ### 21. \(6x^{2} + 7x - 5\) **Factored Form:** \((2x - 1)(3x + 5)\) **Steps:** 1. Multiply the coefficient of \(x^2\) and the constant term: \(6 \times (-5) = -30\). 2. Find two numbers that multiply to -30 and add to 7. These numbers are 10 and -3. 3. Rewrite the middle term: \(6x^2 + 10x - 3x - 5\). 4. Factor by grouping: \[ 2x(3x + 5) - 1(3x + 5) = (2x - 1)(3x + 5) \] --- ### 22. \(8x^{2} - 14x + 3\) **Factored Form:** \((4x - 1)(2x - 3)\) **Steps:** 1. Multiply the coefficient of \(x^2\) and the constant term: \(8 \times 3 = 24\). 2. Find two numbers that multiply to 24 and add to -14. These numbers are -12 and -2. 3. Rewrite the middle term: \(8x^2 - 12x - 2x + 3\). 4. Factor by grouping: \[ 4x(2x - 3) - 1(2x - 3) = (4x - 1)(2x - 3) \] --- Feel free to ask if you need further explanations or assistance with related problems!