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Factor completely: (3x+4)(2x-3)+(5x+2)(2x-3)^(2) Answer:

Factor completely:\newline(3x+4)(2x−3)+(5x+2)(2x−3)2 (3 x+4)(2 x-3)+(5 x+2)(2 x-3)^{2} \newlineAnswer:

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Full solution

Q. Factor completely:\newline(3x+4)(2x−3)+(5x+2)(2x−3)2 (3 x+4)(2 x-3)+(5 x+2)(2 x-3)^{2} \newlineAnswer:
  1. Recognize common factor: Recognize the common factor in the given expression.\newlineThe common factor in both terms is (2x−3)(2x-3). We can factor this out using the distributive property.
  2. Factor out common factor: Factor out the common factor (2x−3)(2x-3).\newlineThe expression becomes (2x−3)[(3x+4)+(5x+2)(2x−3)](2x-3)[(3x+4) + (5x+2)(2x-3)].
  3. Expand second term: Expand the second term inside the brackets.\newlineWe need to multiply (5x+2)(5x+2) by (2x−3)(2x-3) to simplify the expression inside the brackets. This is done by using the distributive property (FOIL method).\newline(5x+2)(2x−3)=5x(2x)+5x(−3)+2(2x)+2(−3)=10x2−15x+4x−6(5x+2)(2x-3) = 5x(2x) + 5x(-3) + 2(2x) + 2(-3) = 10x^2 - 15x + 4x - 6.
  4. Combine like terms: Combine like terms in the expanded expression.\newlineCombine −15x-15x and 4x4x to get −11x-11x.\newlineSo, (5x+2)(2x−3)=10x2−11x−6(5x+2)(2x-3) = 10x^2 - 11x - 6.
  5. Substitute expanded expression: Substitute the expanded expression back into the factored form.\newlineThe expression now reads (2x−3)[(3x+4)+(10x2−11x−6)](2x-3)[(3x+4) + (10x^2 - 11x - 6)].
  6. Distribute across terms: Distribute the (2x−3)(2x-3) across the terms inside the brackets.\newlineWe need to add (3x+4)(3x+4) to (10x2−11x−6)(10x^2 - 11x - 6) before we can distribute (2x−3)(2x-3).\newline(3x+4)+(10x2−11x−6)=10x2−8x−2(3x+4) + (10x^2 - 11x - 6) = 10x^2 - 8x - 2.
  7. Write final factored expression: Write the final factored expression.\newlineThe completely factored form of the expression is (2x−3)(10x2−8x−2)(2x-3)(10x^2 - 8x - 2).

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