### Video Transcript

Find 𝐴𝐵 given that 𝐴 equals
three 𝑎 squared plus five and 𝐵 equals two 𝑎 squared plus six.

We’ve been given two algebraic
expressions, 𝐴 and 𝐵, and asked to find 𝐴𝐵, which is their product. We therefore need to find the
product of three 𝑎 squared plus five and two 𝑎 squared plus six. Each of these algebraic expressions
are binomials. And to find their product, we need
to multiply out the brackets. In doing so, we need to ensure that
we multiply each term in the first binomial by each term in the second. There are a number of different
ways we can do this.

One method is the grid method, in
which we represent the product as a rectangle with sides whose lengths are equal to
the two factors. Because the area of a rectangle is
equal to its length multiplied by its width, the area of the rectangle is equivalent
to the product of the two factors. We can divide the rectangle into
four smaller rectangles and find an expression for the area of each by multiplying
their dimensions.

For the top-left rectangle, three
𝑎 squared multiplied by two 𝑎 squared is six 𝑎 to the fourth power. Remember that when we multiply
powers of the same base together, we add the exponents. For the bottom-left rectangle,
three 𝑎 squared multiplied by six is 18𝑎 squared. So far, we have multiplied the
first term in binomial 𝐴 by each term in binomial 𝐵. Moving to the top right rectangle,
five multiplied by two 𝑎 squared is 10𝑎 squared. And for the final rectangle, five
multiplied by six is 30.

Finally, we find the total area of
the rectangle by summing the expressions for the areas of the four smaller
rectangles, giving six 𝑎 to the fourth power plus 18𝑎 squared plus 10𝑎 squared
plus 30. Simplifying by combining the like
terms in the center of the expression gives six 𝑎 to the fourth power plus 28𝑎
squared plus 30.

So, using the grid method, we’ve
found that 𝐴𝐵, which is the product of the two binomials 𝐴 and 𝐵, is equal to
six 𝑎 to the fourth power plus 28𝑎 squared plus 30.