To find the width of the rectangle, given the area and the length, we can use the formula:
[tex]\[ \text{Area} = \text{Length} \times \text{Width} \][/tex]
Firstly, let's write down the expressions for the area and length of the rectangle:
- Area: [tex]\( A(x) = x^4 + 4x^3 + 3x^2 - 4x - 4 \)[/tex]
- Length: [tex]\( L(x) = x^3 + 5x^2 + 8x + 4 \)[/tex]
Using the formula for the area of a rectangle, we have:
[tex]\[ \text{Width} = \frac{\text{Area}}{\text{Length}} \][/tex]
That is:
[tex]\[ W(x) = \frac{x^4 + 4x^3 + 3x^2 - 4x - 4}{x^3 + 5x^2 + 8x + 4} \][/tex]
When we simplify this rational expression by performing polynomial division, we determine if there is a quotient and remainder. After simplifying, we find that:
[tex]\[ W(x) = x - 1 \][/tex]
Therefore, the width of the rectangle is:
[tex]\[ \boxed{x - 1} \][/tex]
