_4y_____ _ ___3y________

y^2+6y+5 y^2+5y+4

write the q again

As you can see from the appearance above it, it is very hard to line up fractions in this format.

However, your presentation is much better than what we normally encounter.

4y/(y^2 + 6y + 5) - 3y/(y^2 + 5y + 4)
= 4y/((y+1)(y+5)) - 3y/((y+1)(y+4))
= (4y(y+4) - 3y(y+5))/((y+1)(y+4)(y+5))
= (4y^2 + 16 - 3y^2 - 15)/((y+1)(y+4)(y+5))
= (y^2 + 1)/((y+1)(y+4)(y+5))

The given expression is:

(_4y_____)(____3y______)
(y^2 + 6y + 5) (y^2 + 5y + 4)

To simplify this expression, we need to multiply the terms within the brackets. To do this, we use the distributive property, which states that a(b + c) = ab + ac.

Let's multiply the first terms of the brackets:

(_4y_)(y^2 + 5y + 4) = 4y * y^2 + 4y * 5y + 4y * 4

This simplifies to:

4y^3 + 20y^2 + 16y

Now, let's multiply the second terms of the brackets:

(y^2 + 6y + 5)(__3y__)

= (y^2 + 6y + 5) * 3y

= 3y * y^2 + 3y * 6y + 3y * 5

This simplifies to:

3y^3 + 18y^2 + 15y

Now we have simplified both parts of the expression. To obtain the final result, we add the two simplified expressions together:

(4y^3 + 20y^2 + 16y) + (3y^3 + 18y^2 + 15y)

Combining like terms, we get:

7y^3 + 38y^2 + 31y

Therefore, the simplified expression is 7y^3 + 38y^2 + 31y.