Question 378·Hard·Advanced Math

The polynomial $P(x) = 2x^3 + bx^2 + cx + 8$ satisfies the following conditions:

1. When $P(x)$ is divided by $x^2 + 3$, the quotient is $2x + k$ and the remainder is the constant $r$.
2. When $P(x)$ is divided by $x + 1$, the remainder is $5$.

What is the value of $b + c + k + r$?