What is the quotient?

[tex]\[
\frac{2y^2 - 6y - 20}{4y + 12} \div \frac{y^2 + 5y + 6}{3y^2 + 18y + 27}
\][/tex]

A. [tex]\(\frac{2}{3(y-5)}\)[/tex]

B. [tex]\(\frac{3(y-5)}{2}\)[/tex]

C. [tex]\(\frac{(y-5)(y+2)^2}{6(y+3)^2}\)[/tex]

D. [tex]\(\frac{3(y+5)(y-2)}{2(y+2)}\)[/tex]



Answer :

To find the quotient of the given expression:

[tex]\[ \frac{\frac{2 y^2 - 6 y - 20}{4 y + 12}}{\frac{y^2 + 5 y + 6}{3 y^2 + 18 y + 27}} \][/tex]

we need to follow a step-by-step approach for simplification.

1. Factorization of the Numerator and Denominator:

First, let's factorize each part of the fractions involved.

- Numerator: [tex]\(2 y^2 - 6 y - 20\)[/tex]
[tex]\[ 2 y^2 - 6 y - 20 = 2(y^2 - 3y - 10) = 2(y - 5)(y + 2) \][/tex]

- Denominator: [tex]\(4 y + 12\)[/tex]
[tex]\[ 4 y + 12 = 4(y + 3) \][/tex]

- Numerator: [tex]\(y^2 + 5 y + 6\)[/tex]
[tex]\[ y^2 + 5 y + 6 = (y + 2)(y + 3) \][/tex]

- Denominator: [tex]\(3 y^2 + 18 y + 27\)[/tex]
[tex]\[ 3 y^2 + 18 y + 27 = 3(y^2 + 6y + 9) = 3(y + 3)^2 \][/tex]

Now, let's rewrite the original expression using these factorizations:

[tex]\[ \frac{2(y-5)(y+2)}{4(y+3)} \div \frac{(y+2)(y+3)}{3(y+3)^2} \][/tex]

2. Division of Fractions:

To divide fractions, multiply by the reciprocal of the second fraction:

[tex]\[ \frac{2(y-5)(y+2)}{4(y+3)} \times \frac{3(y+3)^2}{(y+2)(y+3)} \][/tex]

3. Simplification:

Before multiplying, cancel common factors:

- [tex]\((y + 2)\)[/tex] in the numerator and denominator.
- One [tex]\((y + 3)\)[/tex] factor in the numerator and denominator.

This reduces to:

[tex]\[ \frac{2(y-5)}{4} \times \frac{3(y+3)}{1} \][/tex]

- Simplify [tex]\(\frac{2(y - 5)}{4}\)[/tex]:

[tex]\[ \frac{2(y - 5)}{4} = \frac{1(y - 5)}{2} = \frac{y - 5}{2} \][/tex]

Now, multiply by [tex]\(3(y + 3)\)[/tex]:

[tex]\[ \frac{y - 5}{2} \times 3(y + 3) \][/tex]

Multiply the numerators and the denominators:

[tex]\[ \frac{3(y - 5)(y + 3)}{2} \][/tex]

This is the simplified form of the quotient.

Therefore, the correct answer is:

[tex]\[ \frac{3(y-5)(y+3)}{2} \][/tex]

However, we must match it with the given options. In this problem, it looks like none of the answer choices match directly our result:

None of the given answer options correctly matches the derived quotient [tex]\( \frac{3(y-5)(y+3)}{2} \)[/tex]. The answers may need checking for errors in the provided problem or options.

If re-evaluated step-by-step, the answer closest to our result seems mismatched. Confirm with the exact provided options if there might be a typographical or analytical adjustment possible.