 So let's look at another way of multiplying two polynomials. A lot of good insight into mathematics can be gained when we switch from one field to another. And it's useful to go back and forth between algebra, which is about numbers and formulas and equations, and geometry, which is about pictures and drawings. So algebraically, when we multiply two polynomials, we apply the distributive property a couple of times. Another way we can multiply is to use a geometric model for multiplication. So remember that the area of a rectangle with a width of a and a height of b is ab. And this means that the product ab can be found by finding the area of a rectangle with width a and height b. For example, let's find the area of a rectangle with width 2x and height x squared. So let's draw a picture. Now we're ultimately going to use this picture to organize our information. So we don't have to worry about getting the relative dimensions correct. We don't have to worry about which of these is larger, as long as one of them is the width and the other one is the height. And the area is going to be the product of width and height. And that gives us an area of 2x cubed. Well, there's not a lot of advantage to just drawing the pictures unless you do something else. The reason this is useful is that we can break a sum into individual pieces. So let's say we want to find the product of 8x times x plus 7. So we can draw a rectangle with width x plus 7 and height 8x. Now we can take this width of x plus 7 and think about it as a width of x and 7. And now we can find the areas of the two rectangles. This first rectangle is 8x high by x wide. So its area will be this second rectangle 8x high by 7 wide. So its area will be 56x. And the area of the whole is going to be the sum of the two bits 8x squared plus 56x. What about x plus 3 times 2x plus 5? Well, we'll draw a picture. Since each factor is a sum of two terms, we'll break each side into two parts. One side is going to be x and 3. And the other side is going to be 2x and 5. And now we have four rectangles whose areas we can calculate. So this first rectangle has width x by height 2x. So its area will be 2x squared. This next rectangle, x by 5, will have area 5x. This next rectangle, 3 by 2x for an area of 6x. And this last rectangle, 3 by 5 for an area of 15. So remember, the area of the rectangle is equal to the product. So we can add up the individual areas to get our product. So our product will be 2x squared plus 5x plus 6x plus 15. And we have some like terms we can combine which will give us the product x plus 3 times 2x plus 5. The important thing to remember is to break each side into as many parts as there are terms in the factor. So here, my first factor has three terms and my second factor has two terms. So I'll break one side of the rectangle into three parts and split the other side into two. So each part corresponds to one term in the factor. So here, my width is going to be x squared, 3x, and 2. And my height is going to be 2x and 5. And now I can find the areas of each of these six rectangles. So we'll get 2x cubed, 5x squared, 6x squared, 15x, 4x, and 10. And if we add up all of these areas, we find the area of the rectangle and so we find the value of the product. So we'll add up our areas 2x cubed, 5x squared, 6x squared, 15x, 4x, and 10 and collect our like terms to get our final answer. One thing that's worth noticing is that if we've set up our terms so they appear in decreasing order, then our like terms will appear on these slanted lines. And that helps us find them so that we can add them. What if one or both of our factors involve a subtraction? Well, here it's useful to remember that any subtraction is the same as adding the additive inverse. So we can rewrite this as 2x plus additive inverse 5 times 3x plus additive inverse 7. And so once again each factor is a sum and so when we draw our picture we can split the sides of the rectangle into parts that correspond to each term of the sum. So if the one side corresponds to the first factor, 2x and negative 5 and the other side corresponds to our other factor, 3x and negative 7. So this first rectangle, 2x by 3x has area 6x squared. This next rectangle is a little weird because it seems it has a width of 2x and a height of negative 7? Well, without worrying about what a height of negative 7 means we can still conclude that the area of this rectangle is the product of width and height 2x by negative 7 which will be negative 14x. And we have a similar situation with this rectangle. The width is negative 5, the height is 3x but if we multiply it with by height we get an area of of minus 15x. And similarly we can calculate the area of this last rectangle with negative 5 by height negative 7 gives us area 35. And the sum of all these pieces gives us the product 2x minus 5 times 3x minus 7. So that'll give us 6x squared minus 14x minus 15x plus 35 and we can collect our like terms and simplify to get our final answer 6x squared minus 29x plus 35.