Let $f(x)=4 x^{2}-3 x$ and $g(x)=x^{2}-x+2$. Find $(f+g)(x),(f-g)(x)$, (fg)(x), and $\left(\frac{f}{g}\right)(x)$. Give the domain of each.

Step 1 ：Let \(f(x)=4 x^{2}-3 x\) and \(g(x)=x^{2}-x+2\). We need to find the sum, difference, product, and quotient of the functions \(f(x)\) and \(g(x)\). The domain of each function is the set of all real numbers for which the function is defined.

Step 2 ：For the sum and difference of the functions, the domain is the intersection of the domains of \(f(x)\) and \(g(x)\). Since both \(f(x)\) and \(g(x)\) are polynomials, their domains are all real numbers. Therefore, the domain of \((f+g)(x)\) and \((f-g)(x)\) is all real numbers.

Step 3 ：For the product of the functions, the domain is also the intersection of the domains of \(f(x)\) and \(g(x)\). Again, since both \(f(x)\) and \(g(x)\) are polynomials, their domains are all real numbers. Therefore, the domain of \((fg)(x)\) is all real numbers.

Step 4 ：For the quotient of the functions, the domain is the set of all real numbers for which the denominator \(g(x)\) is not equal to zero. We need to solve the equation \(g(x) = 0\) to find the values of \(x\) that are not in the domain of \(\left(\frac{f}{g}\right)(x)\).

Step 5 ：\(f+g(x) = 5x^{2} - 4x + 2\) with domain all real numbers.

Step 6 ：\(f-g(x) = 3x^{2} - 2x - 2\) with domain all real numbers.

Step 7 ：\(fg(x) = 4x^{4} - 7x^{3} + 5x^{2} - 6x\) with domain all real numbers.

Step 8 ：\(\left(\frac{f}{g}\right)(x) = \frac{4x^{2} - 3x}{x^{2} - x + 2}\) with domain all real numbers except \(x = \frac{1}{2} - \frac{\sqrt{7}i}{2}\) and \(x = \frac{1}{2} + \frac{\sqrt{7}i}{2}\).

Step 9 ：\(\boxed{\text{The sum of the functions } f(x) \text{ and } g(x) \text{ is } (f+g)(x) = 5x^{2} - 4x + 2 \text{ with domain all real numbers.}}\)

Step 10 ：\(\boxed{\text{The difference of the functions } f(x) \text{ and } g(x) \text{ is } (f-g)(x) = 3x^{2} - 2x - 2 \text{ with domain all real numbers.}}\)

Step 11 ：\(\boxed{\text{The product of the functions } f(x) \text{ and } g(x) \text{ is } (fg)(x) = 4x^{4} - 7x^{3} + 5x^{2} - 6x \text{ with domain all real numbers.}}\)

Step 12 ：\(\boxed{\text{The quotient of the functions } f(x) \text{ and } g(x) \text{ is } \left(\frac{f}{g}\right)(x) = \frac{4x^{2} - 3x}{x^{2} - x + 2} \text{ with domain all real numbers except } x = \frac{1}{2} - \frac{\sqrt{7}i}{2} \text{ and } x = \frac{1}{2} + \frac{\sqrt{7}i}{2}.}\)