What is the simplified form of the quantity 15 x y squared over x squared plus 5x plus 6 all over the quantity 5 x squared y over 2x squared plus 7x plus 3?

A 3y times the quantity 2x plus 1 over x times the quantity x plus 2

B x times the quantity x plus 2 over 3y times the quantity 2x plus 1

C x times the quantity x minus 2 over 3y times the quantity 2x minus 1

D 3y times the quantity 2x minus 1 over x times the quantity x minus 2

A. 3y times the quantity 2x plus 1 over x times the quantity x plus 2

Step-by-step explanation:

We have; 15 x y squared over x squared plus 5x plus 6 all over the quantity 5 x squared y over 2x squared plus 7x plus 3

That is;

15xy² / (x² + 5x + 6) : ((5x²y / (2x²+7x+3))

We can factorize expressions;

x² + 5x + 6 = (x+2) (x+3)

2x²+7x+3 = (2x+1) (x+3)

Therefore we have;

15xy² / (x+2) (x+3) : (5x²y / (2x+1) (x+3))

we can multiply with the reciprocal;

15xy² / (x+2) (x+3) * ((2x+1) (x+3)) / 5x²y

Simplifying;

3y / (x+2) * (2x+1) / x

Multiplying the numerators and then multiplying the denominators.

We get; 3y (2x+1) / x (x+2)

= 3y (2x+1) / x (x+2)

The answer is therefore;

A 3y times the quantity 2x plus 1 over x times the quantity x plus 2