which multiplication expression is equivalent to $\frac{2x^{2}-5x - 3}{4x^{2}+12x + 5}div\frac{3x^{2}-11x +…

which multiplication expression is equivalent to $\frac{2x^{2}-5x - 3}{4x^{2}+12x + 5}div\frac{3x^{2}-11x + 6}{6x^{2}+11x - 10}$?\n$\frac{(x - 3)(2x + 1)}{(2x + 1)(2x + 5)}cdot\frac{(x - 3)(3x - 2)}{(2x + 5)(3x - 2)}$\n$\frac{(2x + 1)(2x + 5)}{(x - 3)(2x + 1)}cdot\frac{(2x + 5)(3x - 2)}{(x - 3)(3x - 2)}$\n$\frac{(x - 3)(2x + 1)}{(2x + 1)(2x + 5)}cdot\frac{(2x + 5)(3x - 2)}{(x - 3)(3x - 2)}$

which multiplication expression is equivalent to $\frac{2x^{2}-5x - 3}{4x^{2}+12x + 5}div\frac{3x^{2}-11x + 6}{6x^{2}+11x - 10}$?\n$\frac{(x - 3)(2x + 1)}{(2x + 1)(2x + 5)}cdot\frac{(x - 3)(3x - 2)}{(2x + 5)(3x - 2)}$\n$\frac{(2x + 1)(2x + 5)}{(x - 3)(2x + 1)}cdot\frac{(2x + 5)(3x - 2)}{(x - 3)(3x - 2)}$\n$\frac{(x - 3)(2x + 1)}{(2x + 1)(2x + 5)}cdot\frac{(2x + 5)(3x - 2)}{(x - 3)(3x - 2)}$

Answer

Explanation:

Step1: Recall division - to - multiplication rule

Dividing by a fraction is equivalent to multiplying by its reciprocal. So, $\frac{2x^{2}-5x - 3}{4x^{2}+12x + 5}\div\frac{3x^{2}-11x + 6}{6x^{2}+11x - 10}=\frac{2x^{2}-5x - 3}{4x^{2}+12x + 5}\times\frac{6x^{2}+11x - 10}{3x^{2}-11x + 6}$.

Step2: Factor the quadratic expressions

  • Factor $2x^{2}-5x - 3=(x - 3)(2x+1)$.
  • Factor $4x^{2}+12x + 5=(2x + 1)(2x+5)$.
  • Factor $6x^{2}+11x - 10=(2x + 5)(3x-2)$.
  • Factor $3x^{2}-11x + 6=(x - 3)(3x - 2)$.

Step3: Substitute the factored forms

Substitute the factored - forms into the multiplication expression: $\frac{(x - 3)(2x+1)}{(2x + 1)(2x+5)}\times\frac{(2x + 5)(3x-2)}{(x - 3)(3x - 2)}$.

Answer:

$\frac{(x - 3)(2x+1)}{(2x + 1)(2x+5)}\cdot\frac{(2x + 5)(3x-2)}{(x - 3)(3x - 2)}$ (the third option)