For the functions \( f(x)=6-x^{2} \) and \( g(x)=x^{2}+4 x-32 \), find \( f+g, f-g \), \( f g \), and \( \frac{f}{g} \). Determine the domain for each function. \( (f-g)(x)=-2 x^{2}-4 x+38 \) (Simplify your answer.) What is the domain of \( f-g \) ? Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The domain of \( f-g \) is \( \} \). (Use a comma to separate answers as needed.) B. The domain of \( f-g \) is \( (-\infty, \infty) \). (Type your answer in interval nçitation.) C. The domain of \( f-g \) is \( \varnothing \). \( (f \cdot g)(x)=\square \) (Simplify your answer.)

To find \( f+g \), \( f-g \), \( fg \), and \( \frac{f}{g} \) for the functions \( f(x) = 6 - x^2 \) and \( g(x) = x^2 + 4x - 32 \): 1. **Sum**: \[ f+g = (6 - x^2) + (x^2 + 4x - 32) = 4x - 26 \] 2. **Difference**: \[ f-g = (6 - x^2) - (x^2 + 4x - 32) = 6 - x^2 - x^2 - 4x + 32 = -2x^2 - 4x + 38 \] 3. **Product**: \[ fg = (6 - x^2)(x^2 + 4x - 32) \] Distributing gives us: \[ fg = 6x^2 + 24x - 192 - x^4 - 4x^3 + 32x^2 \] Combining like terms yields: \[ fg = -x^4 - 4x^3 + 38x^2 - 192 \] 4. **Quotient**: \[ \frac{f}{g} = \frac{6 - x^2}{x^2 + 4x - 32} \] Next, let's evaluate the domain of the difference function \( f-g \). Since both \( f(x) \) and \( g(x) \) are polynomials, they are defined for all real numbers. Thus, the domain of \( f-g \) is: B. The domain of \( f-g \) is \( (-\infty, \infty) \). And for the product \( (f \cdot g)(x) \), we have: \[ (f \cdot g)(x) = -x^4 - 4x^3 + 38x^2 - 192. \]