15x^3(x+y)^2 - 30x^2(x+y) - 45x(x+y)^4
|               |            |
15             30           45 

Look at the numbers: 15, 30, 45

What is the largest integer that will divide evenly 
into all three of those numbers?  It's 15.  So 15 is 
a common factor that can be factored out.

So we know the factoring starts like this:

15      [                        ]

Next we look at the powers of x: x^3, x^2, x^4

15x^3(x+y)^2 - 30x^2(x+y) - 45x(x+y)^4
   |              |           |
  x^3            x^2          x 

What is the largest power of x that will divide evenly into 
all three of those?  x means x^1.  The largest power of x
that will divide evenly into all three of those is x^1 or just
x, the smallest power of x.  So we can factor out x. So the 
factoring will look like this:

15x     [                        ]

Next we look at (x+y)^2, (x+y), and (x+y)^4

Next we look at the powers of x: x^3, x^2, x^4

15x^3(x+y)^2 - 30x^2(x+y) - 45x(x+y)^4
       |              |          |
     (x+y)^2        (x+y)      (x+y)^4

What is the largest power of (x+y) that will divide evenly into 
all three of those?  (x+y) means (x+y)^1.  The largest power of 
(x+y) that will divide evenly into all three of those is (x+y)^1 
or just (x+y), the smallest power of (x+y).  So we can factor out 
(x+y). So the factoring will look like this:

15x(x+y)[                        ]

The first term of the original expression is 15x^3(x+y)^2.
What factors does 15x^3(x+y)^2 have that 15x(x+y) does NOT
have? 15x^3(x+y)^2 has factors x^2 and (x+y) that 15x(x+y)
does not have, because x needs to be multiplied by x^2 to
get x^3 and (x+y) needs to be multiplied by itself (x+y) to
get (x+y)^2.  So the first term to put inside the brackets
is x^2(x+y) 

15x(x+y)[x^2(x+y)                ]

The second term of the original expression is - 30x^2(x+y).
What factors does - 30x^2(x+y) have that 15x(x+y) does NOT
have? - 30x^2(x+y) has only factors -2 and x that 15x(x+y)
does not have, because 15 needs to be multiplied by -2 to
get -30 and x needs to be multiplied by itself x to
get x^2.  So the second term to put inside the brackets
is - 2x.  So now we have 

15x(x+y)[x^2(x+y) - 2x           ]
    

The third and final term of the original expression is - 45x(x+y)^4.
What factors does - 45x(x+y)^4 have that 15x(x+y) does NOT
have? - 45x(x+y)^4 has factors -3 and (x+y)^3
does not have, because 15 needs to be multiplied by -3 to
get -45 and (x+y) needs to be multiplied by (x+y)^2 to
get (x+y)^3.  So the third term to put inside the brackets
is - 2(x+y)^2.  So now we have 

15x(x+y)[x^2(x+y) - 2x - 3(x+y)^3]

Now we have to remove the parentheses within the brackets:

15x(x+y)[x^3 + x^2y - 2x - 3(x+y)(x+y)(x+y)]

15x(x+y)[x^3 + x^2y - 2x - 3(x+y)(x^2+2xy+y^2)]

15x(x+y)[x^3 + x^2y - 2x - 3(x^3+2x^2y+xy^2+x^2y+2xy^2+y^3)]

15x(x+y)[x^3 + x^2y - 2x - 3(x^3+3x^2y+3xy^2+y^3)]

15x(x+y)[x^3 + x^2y - 2x - 3x^3 - 9x^2y - 9xy^2 - 3y^3]

15x(x+y)[-2x^3 - 8x^2y - 9x^2y - 2x - 3y^3]

Since all those terms are negative we can take a negative sign
out front:

-15x(x+y)[2x^3 + 8x^2y + 9xy^2 + 2x + 3y^3]

We can only change the brackets to parentheses
because they no longer hold parentheses:

-15x(x+y)(2x^3 + 8x^2y + 9xy^2 + 2x + 3y^3)

Edwin