what is the quotient?\n$\frac{2y^{2}-6y - 20}{4y + 12}div\frac{y^{2}+5y + 6}{3y^{2}+18y + 27}$\n$\frac{2}{3(y…

what is the quotient?\n$\frac{2y^{2}-6y - 20}{4y + 12}div\frac{y^{2}+5y + 6}{3y^{2}+18y + 27}$\n$\frac{2}{3(y - 5)}$\n$\frac{3(y - 5)}{2}$\n$\frac{(y - 5)(y + 2)^{2}}{6(y + 3)^{2}}$\n$\frac{3(y + 5)(y - 2)}{2(y + 2)}$

what is the quotient?\n$\frac{2y^{2}-6y - 20}{4y + 12}div\frac{y^{2}+5y + 6}{3y^{2}+18y + 27}$\n$\frac{2}{3(y - 5)}$\n$\frac{3(y - 5)}{2}$\n$\frac{(y - 5)(y + 2)^{2}}{6(y + 3)^{2}}$\n$\frac{3(y + 5)(y - 2)}{2(y + 2)}$

Answer

Explanation:

Step1: Factor the polynomials

  • Factor (2y^{2}-6y - 20=2(y^{2}-3y - 10)=2(y - 5)(y+2))
  • Factor (4y + 12=4(y + 3))
  • Factor (y^{2}+5y + 6=(y + 2)(y+3))
  • Factor (3y^{2}+18y + 27=3(y^{2}+6y + 9)=3(y + 3)^{2})

Step2: Rewrite the division as multiplication by reciprocal

The expression (\frac{2y^{2}-6y - 20}{4y + 12}\div\frac{y^{2}+5y + 6}{3y^{2}+18y + 27}) becomes (\frac{2(y - 5)(y + 2)}{4(y + 3)}\times\frac{3(y + 3)^{2}}{(y + 2)(y + 3)})

Step3: Simplify the expression

Cancel out the common factors:

  • Cancel out a factor of 2 from the first - fraction's numerator and the second - fraction's denominator ((\frac{2}{4}=\frac{1}{2})).
  • Cancel out ((y + 2)) and ((y + 3)) terms. We get (\frac{(y - 5)\times3(y + 3)}{2\times(y + 3)}=\frac{3(y - 5)}{2})

Answer:

(\frac{3(y - 5)}{2})