To determine which of the given expressions produces a quadratic function for the functions [tex]\(a(x) = 2x - 4\)[/tex] and [tex]\(b(x) = x + 2\)[/tex], let's evaluate each of the possible expressions step by step.
### Expression 1: [tex]\(\left(\frac{a}{b}\right)(x)\)[/tex]
First, we find the expression [tex]\(\left(\frac{a}{b}\right)(x)\)[/tex]:
[tex]\[ \left(\frac{a}{b}\right)(x) = \frac{a(x)}{b(x)} = \frac{2x - 4}{x + 2} \][/tex]
This is a rational function, not a quadratic function. A quadratic function has the form [tex]\(ax^2 + bx + c\)[/tex], which is a polynomial of degree 2. The rational function [tex]\(\frac{2x - 4}{x + 2}\)[/tex] does not simplify to a quadratic form.
### Expression 2: [tex]\((a - b)(x)\)[/tex]
Next, we find the expression [tex]\((a - b)(x)\)[/tex]:
[tex]\[ (a - b)(x) = a(x) - b(x) = (2x - 4) - (x + 2) = 2x - 4 - x - 2 = x - 6 \][/tex]
This is a linear function, [tex]\(x - 6\)[/tex], which is a polynomial of degree 1. Therefore, it is not a quadratic function.
### Expression 3: [tex]\((a + b)(x)\)[/tex]
Finally, we find the expression [tex]\((a + b)(x)\)[/tex]:
[tex]\[ (a + b)(x) = a(x) + b(x) = (2x - 4) + (x + 2) = 2x - 4 + x + 2 = 3x - 2 \][/tex]
This is also a linear function, [tex]\(3x - 2\)[/tex], which is a polynomial of degree 1. Therefore, it is not a quadratic function.
### Conclusion:
None of the given expressions, [tex]\(\left(\frac{a}{b}\right)(x)\)[/tex], [tex]\((a - b)(x)\)[/tex], or [tex]\((a + b)(x)\)[/tex], produce a quadratic function. Thus, the result is that none of these expressions are quadratic.
So the answer is:
[tex]\[ \boxed{\text{None}} \][/tex]
