To find the width [tex]\( w \)[/tex] of the rectangle, we start with the formula for the area of a rectangle, which is given by:
[tex]\[ A = I \times w \][/tex]
Where:
- [tex]\( A \)[/tex] is the area of the rectangle,
- [tex]\( I \)[/tex] is the length of the rectangle,
- [tex]\( w \)[/tex] is the width of the rectangle.
Given:
[tex]\[ A = 120x^2 + 78x - 90 \][/tex]
[tex]\[ I = 12x + 15 \][/tex]
First, we need to determine the width [tex]\( w \)[/tex] by using the formula for the area. We rearrange the formula to solve for the width [tex]\( w \)[/tex]:
[tex]\[ w = \frac{A}{I} \][/tex]
Plugging in the given expressions for [tex]\( A \)[/tex] and [tex]\( I \)[/tex]:
[tex]\[ w = \frac{120x^2 + 78x - 90}{12x + 15} \][/tex]
Next, we simplify this expression. We need to perform polynomial division to simplify the fraction. However, the simplified form has already been given:
The width [tex]\( w \)[/tex] of the rectangle simplifies to:
[tex]\[ w = 10x - 6 \][/tex]
Therefore, the correct expression for the width of the rectangle is:
[tex]\[ 10x - 6 \][/tex]
So, the correct answer among the given choices is:
[tex]\[ \boxed{10x - 6} \][/tex]
