Question

What is the quotient 4-x/x^2+5x-6 divided by x^2-11x+28/x^2+7x+6 in simplified form?

Answer 1

$\dfrac{4-x}{x^2+5x-6}:\dfrac{x^2-11x+28}{x^2+7x+6}=(*)$

$4-x=-(x-4)\\\\x^2+5x-6=x^2+6x-x-6=x(x+6)-1(x+6)=(x+6)(x-1)\\\\x^2-11x+28=x^2-7x-4x+28=x(x-7)-4(x-7)\\=(x-7)(x-4)\\\\x^2+7x+6=x^2+6x+x+6=x(x+6)+1(x+6)=(x+6)(x+1)\\\\(*)=\dfrac{-(x-4)}{(x+6)(x-1)}:\dfrac{(x-7)(x-4)}{(x+6)(x+1)}$
$=\dfrac{-(x-4)}{(x+6)(x-1)}\cdot\dfrac{(x+6)(x+1)}{(x-7)(x-4)}=\dfrac{-1}{x-1}\cdot\dfrac{x+1}{x-7}\\\\=-\dfrac{x+1}{x^2-7x-x+7}=-\dfrac{x+1}{x^2-8x+7}$