\( (f+g)(x)=5 x^{2}+x+1 \) (Simplify your answer.) What is the domain of \( f+g \) ? A. The domain of \( f+g \) is \( (-\infty, \infty) \). (Type your answer in interval notation.) B. The domain of \( f+g \) is \( \{\quad\} \). (Use a comma to separate answers as needed.) C. The domain of \( f+g \) is \( \varnothing \). \( (f-g)(x)=\square \) (Simplify your answer.)

The expression \( (f+g)(x) = 5x^2 + x + 1 \) is a polynomial, and polynomials have a domain of all real numbers. So, the domain of \( f+g \) is indeed \( (-\infty, \infty) \). Now, moving to the expression for \( (f-g)(x) \): Since \( (f+g)(x) = 5x^2 + x + 1 \), to find \( (f-g)(x) \), we need to express \( f(x) \) and \( g(x) \). We can assume: - Let \( f(x) = 5x^2 + a \) - Let \( g(x) = x + 1 - a \) where \( a \) is some constant. Then, \( (f-g)(x) = (f(x) - g(x)) \) simplifies to: \[ (f-g)(x) = (5x^2 + a) - (x + 1 - a) = 5x^2 - x + 2a - 1. \] If we further define \( a \) to be 0 for simplicity, then \( (f-g)(x) = 5x^2 - x - 1 \) would be the simplified form. Now, for a little fun fact: Did you know that the quadratic equations illustrated here have all sorts of applications, from calculating trajectories in physics to optimizing areas in economics? You can find all sorts of real-world uses for polynomials! As for more problems like this, grab a math puzzle book or an online resource! Those can be fantastic for sharpening your skills and having fun while you're at it. Mathematical concepts can be beautifully complex, and solving more problems can deepen your understanding. Enjoy your math journey!