Let's find out the total number of cubbies Erik can make in the bookshelf. Given:
- The height of the bookshelf: [tex]\(x^2 - 5x - 6\)[/tex]
- The width of the bookshelf: [tex]\(x^2 + 4x + 3\)[/tex]
- The height of each cubby: [tex]\(x + 3\)[/tex]
- The width of each cubby: [tex]\(x - 6\)[/tex]
First, we need to determine the total area of the bookshelf. This is found by multiplying the height and width of the entire bookshelf:
[tex]\[
(x^2 - 5x - 6)(x^2 + 4x + 3)
\][/tex]
Next, we determine the area of one cubby by multiplying its height and width:
[tex]\[
(x + 3)(x - 6)
\][/tex]
To find the total number of cubbies, we must divide the total area of the bookshelf by the area of one cubby:
[tex]\[
\frac{(x^2 - 5x - 6)(x^2 + 4x + 3)}{(x + 3)(x - 6)}
\][/tex]
By expanding and simplifying the expressions, we arrive at:
1. Expand [tex]\(x^2 - 5x - 6\)[/tex]:
[tex]\[
(x^2 - 5x - 6) = (x+1)(x-6)
\][/tex]
2. Expand [tex]\(x^2 + 4x + 3\)[/tex]:
[tex]\[
(x^2 + 4x + 3) = (x+1)(x+3)
\][/tex]
So,
[tex]\[
(x^2 - 5x - 6)(x^2 + 4x + 3) = (x+1)(x-6)(x+1)(x+3)
\][/tex]
3. Expand [tex]\(x + 3\)[/tex] and [tex]\(x - 6\)[/tex]:
[tex]\[
(x + 3)(x - 6)
\][/tex]
Thus,
[tex]\[
\frac{(x+1)(x-6)(x+1)(x+3)}{(x + 3)(x - 6)}
\][/tex]
Upon canceling the common factors in the numerator and the denominator, we obtain:
[tex]\[
(x+1)(x+1) = (x+1)^2
\][/tex]
The simplified expression for the total number of cubbies in Erik's bookshelf is:
[tex]\[
x^2 + 2x + 1
\][/tex]
