Given \( f(x)=2 x^{3}+7 x^{2}-7 x-30 \) and \( g(x)=x-2 \), find \( \left(\frac{f}{g}\right)(x) \)

To find \(\left(\frac{f}{g}\right)(x)\), we need to divide the function \( f(x) \) by \( g(x) \). We have: \[ f(x) = 2x^3 + 7x^2 - 7x - 30 \] and \[ g(x) = x - 2. \] We will perform polynomial long division. 1. Divide the leading term of \( f(x) \) (which is \( 2x^3 \)) by the leading term of \( g(x) \) (which is \( x \)). This gives \( 2x^2 \). 2. Multiply \( g(x) \) by \( 2x^2 \): \[ 2x^2(x - 2) = 2x^3 - 4x^2. \] 3. Subtract this from \( f(x) \): \[ (2x^3 + 7x^2 - 7x - 30) - (2x^3 - 4x^2) = (7x^2 + 4x^2) - 7x - 30 = 11x^2 - 7x - 30. \] 4. Now repeat the process with the new polynomial \( 11x^2 - 7x - 30 \). Divide the leading term \( 11x^2 \) by \( x \) to get \( 11x \). 5. Multiply \( g(x) \) by \( 11x \): \[ 11x(x - 2) = 11x^2 - 22x. \] 6. Subtract: \[ (11x^2 - 7x - 30) - (11x^2 - 22x) = (-7x + 22x) - 30 = 15x - 30. \] 7. Now divide \( 15x - 30 \) by \( g(x) \). Divide the leading term \( 15x \) by \( x \) to get \( 15 \). 8. Multiply \( g(x) \) by \( 15 \): \[ 15(x - 2) = 15x - 30. \] 9. Subtract: \[ (15x - 30) - (15x - 30) = 0. \] Since the remainder is 0, we have completed the division. So, we find: \[ \frac{f(x)}{g(x)} = 2x^2 + 11x + 15. \] Thus, the answer is: \[ \left(\frac{f}{g}\right)(x) = 2x^2 + 11x + 15. \]