Determine the values of the constants \( k \) and \( r \), given that when \( p(x) \) is divided by \( (x+1) \) the remainder is \( r \) and when \( p(x) \) is divided by \( (x-2) \) the remainder is \( 2 r \), where \( p(x)=x^{3}-k x^{2}+2 x+12 \).

To determine the values of the constants \( k \) and \( r \), we start by applying the Remainder Theorem. According to the theorem, the remainder of a polynomial \( p(x) \) when divided by \( (x - a) \) is simply \( p(a) \). 1. **Finding the remainder when divided by \( (x + 1) \)**: To find \( r \), we need to evaluate \( p(-1) \): \[ p(-1) = (-1)^3 - k(-1)^2 + 2(-1) + 12 = -1 - k - 2 + 12 = 9 - k. \] Therefore, we have: \[ r = 9 - k. \tag{1} \] 2. **Finding the remainder when divided by \( (x - 2) \)**: We find \( 2r \) by evaluating \( p(2) \): \[ p(2) = (2)^3 - k(2)^2 + 2(2) + 12 = 8 - 4k + 4 + 12 = 24 - 4k. \] Since the remainder is \( 2r \), we have: \[ 2r = 24 - 4k. \tag{2} \] 3. **Solving the equations**: We now have two equations, (1) and (2): \[ r = 9 - k, \quad \text{and} \quad 2r = 24 - 4k. \] Substituting equation (1) into equation (2): \[ 2(9 - k) = 24 - 4k. \] This simplifies to: \[ 18 - 2k = 24 - 4k. \] Rearranging gives: \[ 4k - 2k = 24 - 18, \] \[ 2k = 6 \quad \Rightarrow \quad k = 3. \] 4. **Finding \( r \)**: Now we substitute \( k = 3 \) back into equation (1): \[ r = 9 - k = 9 - 3 = 6. \] Thus, the values of the constants are: \[ \boxed{k = 3} \quad \text{and} \quad \boxed{r = 6}. \]