To simplify the expression, we first factorize the expressions in the numerator and the denominator:

Factorize the numerator:
4y^2 + 10y - 7 = (4y - 1)(y + 7)

Factorize the denominator:
16y^2 - 9 = (4y + 3)(4y - 3)

Now substitute the factorized expressions into the original expression:

[(4y - 1)(y + 7)] / [(4y + 3)(4y - 3)] = [(3y - 7) / 3 - 4y] + [(6y + 5) / 3 + 4y]

Now we simplify the fractions separately:

For the first fraction:
(3y - 7) / 3 - 4y = (3y - 7 - 12y) / 3 = (-9y - 7) / 3 = -3y - 7/3

For the second fraction:
(6y + 5) / 3 + 4y = (6y + 5 + 12y) / 3 = (18y + 5) / 3 = 6y + 5/3

Therefore, the simplified expression is:
-3y - 7/3 + 6y + 5/3