One simple way to find the LCM for the denominators of two fractions is to multiply the denominators together. If we do that, we can use FOIL (first, outer, inner, last) to expand the terms in parentheses, and we have. . .
(7y - 3) * (7y + 3) = 49y^2 + 21y - 21y - 9
which reduces to
49y^2 - 9
But if we change the denominators, we must also change the numerators, so that their proportion is not changed. A simple way to do that is to multiply each fraction by 1 in a special way. Starting with the first fraction 4 / (7y - 3), we can multiply it by . . .
(7y + 3) / (7y + 3)
which is equal to 1, since any fraction with the same numerator and denominator is equal to 1.
Then. . .
(4 * (7y + 3)) / ((7y + 3) * (7y - 3))
This denominator is just the LCM that we calculated before, which is 49y^2 - 9. Multiplying out the numerator, we have
(28y + 12) / (49y^2 - 9)
Repeating this process for the other fraction, but now multiplying by 1 written as (7y - 3) / (7y - 3) we get. . .
(-5 * (7y - 3)) / ((7y + 3) * (7y - 3))
Which is the same as
(-35y + 15) / (49y^2 - 9).