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

`(4-x)/(x^2+5x-6):(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)\\\\(*)=(-(x-4))/((x+6)(x-1)):((x-7)(x-4))/((x+6)(x+1))`

`=(-(x-4))/((x+6)(x-1))\cdot((x+6)(x+1))/((x-7)(x-4))=(-1)/(x-1)\cdot(x+1)/(x-7)\\\\=-(x+1)/(x^2-7x-x+7)=-(x+1)/(x^2-8x+7)`