Show all work in dividing the question below

Question

Show all work in dividing the quantity y-squared minus 3y plus 2 all over the quantity y-squared plus 5y plus 6 by the quantity y-squared plus 2y minus 8 all over the quantity y-squared plus y minus 12 . Use complete sentences to explain how to simplify this expression.

Answer 1

First, fully factor each numerator and denominator:

 

y² - 3y + 2 = (y - 2)(y - 1)

y² + 5y  + 6 = (y + 3)(y + 2)

y² + 2y - 8 = (y + 4)(y - 2)

y² + y - 12 = (y - 3)(y + 4)

 

Now, instead of dividing by the second expression, multiply by its reciprocal:

 

((y - 2)(y - 1))/((y + 3)(y + 2)) × ((y - 3)(y + 4))/((y + 4)(y - 2))

 

Finally, cross out duplicate factors in the top and bottom and combine the fractions:

 

((y - 1)(y - 3)) / ((y + 2)(y + 3))