Multiplying two polynomials means multiplying each term in the first expression by each term in the second expression, and then adding them all together. Let's take it just a little bit at a time.
First we will multiply A by the second expression, but we'll do this one term at a time.
A * A3 = A4 Remember that multiplying similar variables is the same as adding exponents. A1 + A3 = A4 because 1 + 3 = 4
A * -3AB = -3A2B Multiplying different variables is the same as multiplying a variable and a constant. 5 * x is as simple as that expression can get, so it just stays at 5x (we take out the multiplication symbol to make it easier to read. It's still there though).
A * -b2 = -AB2 This one is the same thing as the second one. Remember that one negative in an expression will make the entire expression negative, which is why we can move the - symbol to the front.
Now we need to add these all together. Using the same rules, we come up with A4 - 3A2B - AB2
B * A3 = A3B
B * -3AB = -3AB2