Firstly, we should note the first 6 rows of Pascal's triangle (for more information see the Wikipedia entry on Pascal's triangle).

              1             
             1 1
            1 2 1
           1 3 3 1
          1 4 6 4 1
         1 5 x x 5 1

using x=10 (for neatness)

Using polynomial expansion, which employs Pascal's trianagle, we would obtain for the (y+3)³ component:

y³ + 9y² + 27y + 27 (1)

and for the (2-y)⁵ component:

-y⁵ + 10y⁴ -40y³ + 80y² -80y + 32 (2)

The point Colin was making was that to obtain the co-efficient of y⁴ after components (1) and (2) multiply is that you are interested in the products of:

y³ and -80y

9y² and 80y²

27y and -40y³

27 and 10y⁴

You can simply consider addition of the products of the coefficients for each of the four pairs:

-80 + (9 x 80) + (27 x -40) + (27 x 10) = -80 + 720 -1080 + 270 = -170

Thus, co-efficient of y⁴ in the expansion of (y+3)³(2-y)⁵ is -170.

This is really easy to complete with the Binomial Theorem (example)

The procedure is to find the y⁴ term, which can either be:

1. y³ x y¹
2. y² x y²
3. y¹ x y³
4. y⁰ x y⁴

All these terms will exist, and will create a y⁴ term, so (i) find each of their coefficients and then (ii) add them together.

Let me know if you need more guidance than that.