### Video Transcript

π΄π΅πΊπ» is a rectangle. π΅πΆπ·πΊ is a rectangle. πΊπ·πΈπΉ is a square. These three shapes join together to
make a shape, π
. Show that the total area of π
is
four π₯ squared plus eight π₯ plus five.

Our first step is to work out the
dimensions of each of the three shapes. π΄π΅πΊπ» is a rectangle. It has length π₯ plus two
centimeters and width four centimeters. π΅πΆπ·πΊ is also a rectangle. It has length two π₯ minus one
centimeters and width four centimeters. This is because the lengths of π΅πΆ
and πΉπΈ are equal. Likewise, π΄π» is equal to
πΆπ·. Finally, πΊπ·πΈπΉ is a square. All four lengths of a square are
equal. In this case, they are two π₯ minus
one centimeters.

We need to work out an expression
for the area of each of these three shapes and then add them together to make the
area of shape π
. The area of a rectangle is
calculated by multiplying the length by the width this means that the area of
π΄π΅πΊπ» is equal to four multiplied by π₯ plus two. In the same way, the area of the
rectangle π΅πΆπ·πΊ is equal to four multiplied by two π₯ minus one.

The area of a square can be
calculated by squaring its length. In this case, we have two π₯ minus
one all squared. The total area of π
is therefore
equal to two π₯ minus one all squared plus four multiplied by π₯ plus two plus four
multiplied by two π₯ minus one.

Our next step is to multiply out or
expand the brackets. Expanding a single bracket involves
multiplying the number outside the bracket by each of the terms inside the
bracket. If we look at the first rectangle,
we need to multiply four by π₯ and four by two. Four multiplied by π₯ is equal to
four π₯ and four multiplied by two is equal to eight. Therefore, the area of rectangle
π΄π΅πΊπ» is four π₯ plus eight.

We can use the same method to
expand four multiplied by two π₯ minus one. Four multiplied by two π₯ is equal
to eight π₯ and four multiplied by negative one is equal to negative four. This means that the area of the
rectangle π΅πΆπ·πΊ is eight π₯ minus four.

Finally, we need to expand two π₯
minus one all squared. Squaring an expression means
multiplying it by itself. In this case, we need to multiply
two π₯ minus one by two π₯ minus one. We will look at two methods of
expanding double brackets: the FOIL method and the grid method.

The FOIL method involves
multiplying the First terms, the Outside terms, the Inside terms, and the Last
terms. Multiplying the first terms two π₯
and two π₯ gives us four π₯ squared. Multiplying the outside terms two
π₯ and negative one gives us negative two π₯. Multiplying the inside terms also
gives us negative two π₯. Finally, multiplying the last terms
gives us positive one. Negative one multiplied by negative
one is equal to positive one. Remember that when you multiply two
negative numbers, you get a positive answer.

We can now group or collect the
like terms: negative two π₯ minus two π₯. This is equal to negative four
π₯. We can therefore see that two π₯
minus one all squared or two π₯ minus one multiplied by two π₯ minus one is equal to
four π₯ squared minus four π₯ plus one.

An alternative method to expand
these brackets is the grid method. Once again, we need to multiply
each term in the first bracket by each term in the second bracket. Two π₯ multiplied by two π₯ is four
π₯ squared. Negative one multiplied by two π₯
is equal to negative two π₯. Two π₯ multiplied by negative one
is also equal to negative two π₯. Negative one multiplied by negative
one is equal to positive one. Once again, we have two negative
numbers multiplying to give a positive answer. Grouping the two negative two π₯s
gives us negative four π₯. Once again, we have proved that the
area of the square is equal to four π₯ squared minus four π₯ plus one.

The total area of π
is therefore
equal to four π₯ squared minus four π₯ plus one plus four π₯ plus eight plus eight
π₯ minus four. We now need to collect the like
terms. We can group negative four π₯,
positive four π₯, and positive eight π₯ and we can also group positive one, positive
eight, and negative four. There is only one term with π₯
squared. Therefore, our first term is four
π₯ squared. Negative four π₯ plus four π₯ is
equal to zero. Adding eight π₯ to this gives us
positive eight π₯. One plus eight is equal to
nine. Subtracting four gives us five.

This means that the total area of
π
is four π₯ squared plus eight π₯ plus five as required in the question.