enter the correct answer in the box.\nwhat is the simplest form of the expression representing this…

enter the correct answer in the box.\nwhat is the simplest form of the expression representing this product?\n\\( \\frac { x + 10 } { x ^ { 2 } + 7 x - 18 } \\cdot \\frac { 3 x ^ { 2 } - 12 x + 12 } { 3 x + 30 } \\)

enter the correct answer in the box.\nwhat is the simplest form of the expression representing this product?\n\\( \\frac { x + 10 } { x ^ { 2 } + 7 x - 18 } \\cdot \\frac { 3 x ^ { 2 } - 12 x + 12 } { 3 x + 30 } \\)

Answer

Explanation:

Step1: Factor the expressions

  • Factor (x^{2}+7x - 18): (x^{2}+7x - 18=(x + 9)(x-2)) (using (x^{2}+(a + b)x+ab=(x + a)(x + b)) where (a = 9), (b=-2) and (ab=-18), (a + b=7)).
  • Factor (3x^{2}-12x + 12): (3x^{2}-12x + 12=3(x^{2}-4x + 4)=3(x - 2)^{2}) (using (a^{2}-2ab + b^{2}=(a - b)^{2}) with (a=x), (b = 2)).
  • Factor (3x + 30): (3x+30=3(x + 10)).

The product becomes (\frac{x + 10}{(x + 9)(x-2)}\cdot\frac{3(x - 2)^{2}}{3(x + 10)}).

Step2: Cancel out the common factors

Cancel out the common factors:

  • Cancel out (x + 10) (since (\frac{x + 10}{x + 10}=1)).
  • Cancel out (3) (since (\frac{3}{3}=1)).
  • Cancel out one factor of (x - 2) (since (\frac{(x - 2)^{2}}{x - 2}=x - 2)).

We get (\frac{x - 2}{x + 9}).

Answer:

(\frac{x-2}{x + 9})