Answer:
Option (1)
Step-by-step explanation:
The given expression is,
[tex]\frac{x^{2}-19x+90}{x^{2}+19x+90}[/tex] ÷ [tex]\frac{x^{2}-2x-80}{x^{2}-x-72}[/tex]
Now we will factor each polynomial separately and substitute the factors in the expression.
x² - 19x + 90 = x² - 10x - 9x + 90
= x(x - 10) - 9(x - 10)
= (x - 10)(x - 9)
x² - 2x - 80 = x² - 10x + 8x - 80
= x(x - 10) + 8(x - 10)
= (x + 8)(x - 10)
x²- x - 72 = x² - 9x + 8x - 72
= x(x - 9) + 8(x - 9)
= (x + 8)(x - 9)
x² + 19x + 90 = x² + 10x + 9x + 90
= x(x + 10) + 9(x + 10)
= (x + 9)(x + 10)
Now by substituting these factors in the given expression
[tex]\frac{(x - 10)(x - 9)}{(x + 10)(x + 9)}[/tex] ÷ [tex]\frac{(x + 8)(x - 10)}{(x + 8)(x - 9)}[/tex] = [tex]\frac{(x - 10)(x - 9)}{(x + 10)(x + 9)}[/tex] ÷ [tex]\frac{(x - 10)}{(x - 9)}[/tex]
= [tex]\frac{(x - 10)(x - 9)}{(x + 10)(x + 9)}[/tex] × [tex]\frac{(x - 9)}{(x - 10)}[/tex]
= [tex]\frac{(x - 9)^2}{(x + 10)(x + 9)}[/tex]
Therefore, Option (1) will be the answer.
