To find the width of the rectangle given the area and the length, consider the relationship:
[tex]\[ \text{Area} = \text{Length} \times \text{Width} \][/tex]
Let’s denote the area of the rectangle by [tex]\( A \)[/tex], the length by [tex]\( L \)[/tex], and the width by [tex]\( W \)[/tex]:
[tex]\[ A = L \times W \][/tex]
Given the area [tex]\( A \)[/tex] is:
[tex]\[ A = x^4 + 4x^3 + 3x^2 - 4x - 4 \][/tex]
And the length [tex]\( L \)[/tex] is:
[tex]\[ L = x^3 + 5x^2 + 8x + 4 \][/tex]
We can find the width [tex]\( W \)[/tex] by rearranging the relationship to solve for [tex]\( W \)[/tex]:
[tex]\[ W = \frac{A}{L} \][/tex]
Substitute the given expressions for [tex]\( A \)[/tex] and [tex]\( L \)[/tex] into the equation:
[tex]\[ W = \frac{x^4 + 4x^3 + 3x^2 - 4x - 4}{x^3 + 5x^2 + 8x + 4} \][/tex]
Simplify this expression to find [tex]\( W \)[/tex]. After performing the polynomial division, we obtain:
[tex]\[ W = x - 1 \][/tex]
Hence, the width of the rectangle is:
[tex]\[ \boxed{x - 1} \][/tex]
