48x²y + 72xy² + 27y³

The GCF is 3y.  So we take that out:

3y(16x² + 24xy + 9y²)

Notice that the first and last terms of the trinomial in 
the parentheses are squares since 16x² = (4x)² and 9y² = (3y)².
Whenever that is the case, we remember the rule that 

A² + 2AB + B² = (A + B)²

and we check to see if we let A = 4x and B = 3y, then 2AB would equal the
middle term 24xy.  We see that it would, since 2(4x)(3y) does equal to 24xy,
the middle term in the trinomial, so (A + B)² becomes (4x + 3y)² and so

3y(16x² + 24xy + 9y²)

becomes

3y(4x + 3y)²

Edwin