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 * A

^{3 }= A

^{4}Remember that multiplying similar variables is the same as adding exponents. A

^{1 }+ A

^{3 }= A

^{4}because 1 + 3 = 4

A * -3AB = -3A

^{2}B 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 * -b

^{2 }= -AB

^{2}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 A

^{4 }- 3A

^{2}B - AB

^{2}

B * A

^{3}= A

^{3}B

B * -3AB = -3AB

^{2}

^{2 }= -B

^{3}

^{3}B - 3AB

^{2 }- 3B

^{3}

^{4 }+ A

^{3}B - 3A

^{2}B - 4AB

^{2 }- B

^{3}