Sure, let's go step by step to solve this problem.
### Step 1: Define the Heights and Widths
Given:
- Bookshelf height: [tex]\( x^2 - 2x - 8 \)[/tex]
- Bookshelf width: [tex]\( x^2 + 5x + 6 \)[/tex]
- Cubby height: [tex]\( x + 3 \)[/tex]
- Cubby width: [tex]\( x - 4 \)[/tex]
### Step 2: Expression for Total Area of the Bookshelf
The total area of the bookshelf can be expressed by multiplying the height and width of the bookshelf:
[tex]\[ \text{Bookshelf Area} = (x^2 - 2x - 8) \cdot (x^2 + 5x + 6) \][/tex]
### Step 3: Expression for Area of One Cubby
The area of one cubby is given by multiplying the height and width of the cubby:
[tex]\[ \text{Cubby Area} = (x + 3) \cdot (x - 4) \][/tex]
### Step 4: Expression for Total Number of Cubbies
To find the total number of cubbies, divide the total area of the bookshelf by the area of one cubby:
[tex]\[ \text{Total Cubbies} = \frac{(x^2 - 2x - 8) \cdot (x^2 + 5x + 6)}{(x + 3) \cdot (x - 4)} \][/tex]
### Step 5: Simplifying the Expression
First, simplify the numerator and the denominator separately.
#### Numerator:
Factorize [tex]\( x^2 - 2x - 8 \)[/tex] and [tex]\( x^2 + 5x + 6 \)[/tex]:
[tex]\[ x^2 - 2x - 8 = (x - 4)(x + 2) \][/tex]
[tex]\[ x^2 + 5x + 6 = (x + 2)(x + 3) \][/tex]
So, the numerator becomes:
[tex]\[ (x - 4)(x + 2)(x + 2)(x + 3) \][/tex]
#### Denominator:
[tex]\[ (x + 3)(x - 4) \][/tex]
### Step 6: Reducing the Expression
Now, cancel out the common factors in the numerator and the denominator:
[tex]\[ \frac{(x - 4)(x + 2)(x + 2)(x + 3)}{(x + 3)(x - 4)} \][/tex]
Cancelling the common terms [tex]\((x - 4)\)[/tex] and [tex]\((x + 3)\)[/tex], we get:
[tex]\[ \frac{(x + 2)(x + 2)}{1} = (x + 2)^2 \][/tex]
Thus, the simplified expression for the total number of cubbies is:
[tex]\[ x^2 + 4x + 4 \][/tex]
### Final Answer
The total number of cubbies in the entire bookshelf is:
[tex]\[ x^2 + 4x + 4 \][/tex]
