We will start off by factorising both terms into their factors.
z^3 + 6z^2 + 9z
= z(z^2 + 6z + 9)
= z(z + 3)(z + 3)
z^2 - 6z
= z(z - 6)
Hence, the lowest common multiple will be z(z - 6)(z + 3)(z + 3), since z was the only common factor of the two terms besides 1.