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CCSS Math: HSA.APR.A.1, HSA.CED.A.1

We're told that the glass
portion of the window shown below-- that's the window--
has a height to width the ratio of 3:2. So the height of the glass
portion can be represented as 3x and the width of the
glass portion can be represented as 2x. The wood trim adds 7 inches to
the total width and 8 inches to the total height. So let me write that down. So 8 in height, we're
doing it in yellow. So 8 inches to the total height,
I'll do it in yellow, and 7 inches to the total
width, I'll do in this purplish color, right? We're doing the width
in purple. Now what they say is write a
polynomial expression that represents the total area of
the window, including the glass and wood. Simplify the expression. Now, the way they've described
it, they're telling us the ratio of 3:2 for the
glass portion. I think when they're talking
about the glass portion, they're talking about this
entire portion right here. Even though it's not all glass,
you have that thing in the middle, you have this
divider right over here, that's not glass, but I think
they're including that as the glass portion, because they're
not giving us any measurements for that. So the height of the glass
portion can be represented as 3x. So this height right here of
just the glass portion is 3x. You can't see that. Let me do it in a
better color. This portion right here is 3x,
just this distance right here. That is the height of
the glass portion. And then they tell us the wood
trim adds 8 inches to the total height. So if you take that 3x and then
you add that plus that, this whole distance over here,
which is the glass portion plus the wood trim on either
end, it adds 8 inches to the glass portion. The glass portion is 3x, so
we're going to add 8 to it. So this height right
here 3x plus 8. And now let's think
about the width. The width of the glass portion
can be represented as 2x, so that distance right there, just
the glass portion, is 2x. But then they say when you add
the wood trim, the wood trim adds 7 inches to the total
width, so this total width right here, because we're adding
that piece and that piece to it, is 2x plus 7. That is the total width of
the window, the total height is 3x plus 8. Now, what they want us to
do is write a polynomial expression that represents the
total area of the window including the glass and wood. Simplify the expression. Now, the total area is
just going to be the height times the width. So let's just write that down. The area is just going to be
equal to the height, which is 3x plus 8 times the width,
times 2x plus 7. And that is the expression
we've written up. Well, we haven't done
it as a polynomial. We've written it as a product
of two binomials. And if we want to simplify it,
we can just multiply it out. So the way you can think about
it is we multiply-- you're doing the distributive property
twice essentially. But you could say let's multiply
3x times 2x, so you get 3x times 2x. That's multiplying
it times that. Then you want to multiply 3x
times 7, which is 21x-- let me just write it this way--
plus 3x times 7. And then you want to multiply
this 8 times both of these terms. So then you have
plus 8 times 2x. Let me do this. So plus 8 times 2x and then
you have plus 8 times 7. And now we just have to
simplify everything. So what does this become? 3x times 2x is 6x squared plus
3x times 7 is plus 21x plus 8 times 2x, so plus 16x, and then
plus 8 times 7, 8 times 7 is 56, plus 56. We can combine these
two middle terms. 21x plus 16x is what? 37x. So then we get-- let me just
do it in another color-- 6x squared plus 37x plus 56. Now we've expressed it as
a simplified polynomial expression.