If \( a(x)=2 x-4 \) and \( b(x)=x+2 \), which of the following expressions produces a quadratic function? \( (a b)(x) \) \( \left(\frac{a}{b}\right)(x) \) \( (a-b)(x) \) \( (a+b)(x) \)

To determine which expression produces a quadratic function, we need to analyze each option: 1. **For \( (a b)(x) \)**: This expression represents the product of \( a(x) \) and \( b(x) \). Expanding this, we get: \[ (a b)(x) = a(x) \cdot b(x) = (2x - 4)(x + 2) = 2x^2 + 4x - 4x - 8 = 2x^2 - 8, \] which is a quadratic function. 2. **For \( \left(\frac{a}{b}\right)(x) \)**: This expression represents the division of \( a(x) \) by \( b(x) \): \[ \left(\frac{a}{b}\right)(x) = \frac{2x - 4}{x + 2}. \] This expression does not represent a quadratic function; instead, it is a rational function. 3. **For \( (a-b)(x) \)**: This expression represents the difference between \( a(x) \) and \( b(x) \): \[ (a-b)(x) = (2x - 4) - (x + 2) = 2x - 4 - x - 2 = x - 6. \] This is a linear function, not a quadratic. 4. **For \( (a+b)(x) \)**: This expression represents the sum of \( a(x) \) and \( b(x) \): \[ (a+b)(x) = (2x - 4) + (x + 2) = 2x - 4 + x + 2 = 3x - 2. \] This also results in a linear function, not a quadratic. Thus, the expression that produces a quadratic function is \( (a b)(x) \) which yields \( 2x^2 - 8 \).