To find the function [tex]\(a(x)\)[/tex] that models the area of the rectangle, we need to combine the given functions for width [tex]\(w(x)\)[/tex] and length [tex]\(l(x)\)[/tex].
1. The width of the rectangle is given by:
[tex]\[
w(x) = 3x - 6
\][/tex]
2. The length of the rectangle is given by:
[tex]\[
l(x) = 5x + 7
\][/tex]
3. The area of the rectangle [tex]\(a(x)\)[/tex] is found by multiplying the width by the length:
[tex]\[
a(x) = w(x) \times l(x)
\][/tex]
4. Substitute the expressions for [tex]\(w(x)\)[/tex] and [tex]\(l(x)\)[/tex]:
[tex]\[
a(x) = (3x - 6) \times (5x + 7)
\][/tex]
5. Expand the multiplication using the distributive property (also known as FOIL for binomials):
[tex]\[
(3x - 6)(5x + 7) = 3x \cdot 5x + 3x \cdot 7 - 6 \cdot 5x - 6 \cdot 7
\][/tex]
6. Perform the individual multiplications:
[tex]\[
= 15x^2 + 21x - 30x - 42
\][/tex]
7. Combine like terms:
[tex]\[
15x^2 + 21x - 30x - 42 = 15x^2 - 9x - 42
\][/tex]
Therefore, the function that models the area of the rectangle is:
[tex]\[
a(x) = 15x^2 - 9x - 42
\][/tex]
So, the correct answer is:
[tex]\[
\boxed{15x^2 - 9x - 42}
\][/tex]
