The volume of the box is represented by (x^2+5x+6)(x+5). Find

the polynomial that represents the area of the bottom of the box.

can some explain how to get the solution

Divide the volume by the height to get the area of the bottom. You don't say what the height is. You need to know it to do the problem.

the height is x+2

so the problem is
(x^2+5x+6)(x+5)+(x+2)
(x^2+10x^2+31x+30)+x+2
x^3+10^2+32x+30+2
x^3+10x^2+32x+32

Did I do this correct

Your expression factors into
(x+2)(x+3)(x+5)
where the individual factors would represent the width, length, and height of the box in no particular order.

So multiplying any two of the three could be the area of the bottom.
(x+2)(x+3) ---> x^2 + 5x + 6 or
(x+2)(x+5) ---> x^2 + 7x + 10 or
(x+3)(x+5) ---> x^2 + 8x + 15

(the box could be placed with any one of its 3 different faces at the bottom)

If the height is x+2....

(x^2+5x+6)(x+3)/x+2....

Correct?

(x^2+5x+6)=(x+3)(x+2)
(x+3)=(x+3)
So...

(x+3)(x+2)(x+3)/x+2 = (x+3)(x+3)

Is this correct?

No, that is not correct.

When dividing the volume by the height to get the area of the bottom, you need to divide the entire expression representing the volume by the height, which is (x+2).

So, the correct division would look like this:

(x^2+5x+6)(x+5) รท (x+2)

To divide polynomials, you can use long division or synthetic division. In this case, we will use long division.

Start by dividing the first term in the volume expression (x^2) by the first term in the height expression (x). This gives you x.

x
-------------
(x+2) | (x^2+5x+6)(x+5)

Next, multiply the divisor (x+2) by the quotient (x). This gives you (x^2+2x).

x
-------------
(x+2) | (x^2+5x+6)(x+5)
-(x^2+2x)
-------
3x+6

Now subtract this result (3x+6) from the original expression (x^2+5x+6)(x+5).

x+3
-------------
(x+2) | (x^2+5x+6)(x+5)
-(x^2+2x)
-------
3x+6
-(3x+6)
-------
0

Since the remainder is 0, we have successfully divided the volume expression by the height.

Therefore, the polynomial that represents the area of the bottom of the box is the quotient obtained during the division:

x+3

So, the area of the bottom of the box is represented by the polynomial x+3.