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Factor completely: (x+4)(5x-9)+(4x-7)(x+4) Answer:

Factor completely:\newline(x+4)(5x−9)+(4x−7)(x+4) (x+4)(5 x-9)+(4 x-7)(x+4) \newlineAnswer:

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Full solution

Q. Factor completely:\newline(x+4)(5x−9)+(4x−7)(x+4) (x+4)(5 x-9)+(4 x-7)(x+4) \newlineAnswer:
  1. Distribute Terms in Parentheses: Distribute the terms within each set of parentheses.\newlineWe will apply the distributive property a(b+c)=ab+aca(b + c) = ab + ac to both (x+4)(5x−9)(x+4)(5x-9) and (4x−7)(x+4)(4x-7)(x+4).\newlineFirst, for (x+4)(5x−9)(x+4)(5x-9):\newline5x(x)+5x(−9)+4(5x)+4(−9)5x(x) + 5x(-9) + 4(5x) + 4(-9)\newline=5x2−45x+20x−36= 5x^2 - 45x + 20x - 36\newline=5x2−25x−36= 5x^2 - 25x - 36
  2. Distribute Terms for Expressions: Now, distribute the terms for (4x−7)(x+4)(4x-7)(x+4):4x(x)+4x(4)−7(x)−7(4)4x(x) + 4x(4) - 7(x) - 7(4)=4x2+16x−7x−28= 4x^2 + 16x - 7x - 28=4x2+9x−28= 4x^2 + 9x - 28
  3. Combine Like Terms: Add the results from Step 11 and Step 22.\newlineWe combine like terms from the expressions 5x2−25x−365x^2 - 25x - 36 and 4x2+9x−284x^2 + 9x - 28.\newline(5x2−25x−36)+(4x2+9x−28)(5x^2 - 25x - 36) + (4x^2 + 9x - 28)\newline=5x2+4x2−25x+9x−36−28= 5x^2 + 4x^2 - 25x + 9x - 36 - 28\newline=9x2−16x−64= 9x^2 - 16x - 64
  4. Factor by Grouping: Factor by grouping, if possible.\newlineLooking at the expression 9x2−16x−649x^2 - 16x - 64, we can try to factor by grouping. However, since there are no common factors and the trinomial does not factor neatly, we cannot factor by grouping in this case.
  5. Factor Quadratic Expression: Factor the quadratic expression, if possible.\newlineWe look for two numbers that multiply to (9)(−64)=−576(9)(-64) = -576 and add to −16-16. After checking possible factors, we find that there are no such integers that satisfy both conditions. Therefore, the quadratic expression 9x2−16x−649x^2 - 16x - 64 cannot be factored further over the integers.

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