y^2 - 81 = (y - 9) (y + 9)
y^2 + 15y + 54 = (y + 6) (y + 9)
lcm is (y - 9) (y + 9) (y + 6) because (y - 9) (y + 9) (y + 6) contains (y - 9) (y + 9) (to make y^2 - 81) and (y + 6) (y + 9) (to make y^2 + 15y + 54).
the lcm is (y - 9) (y + 9) (y + 6)
or, if you want to expand it
(y^2 - 49) (y + 6)
y ^3 + 6y^2 - 49y - 294