Given P(x) = x2 + bx + 2 and the divisor is x+3, which means a = -3. The remainder is 5.

The Remainder Theorem states that if a polynomial $P(x)$ is divided by $x-a$, the remainder is $P(a)$.
Given $P(x) = x^2 + bx + 2$ and the divisor is $x+3$, which means $a = -3$. The remainder is 5.
Therefore, we can write:

$$P(-3) = 5$$

Substitute $x = -3$ into $P(x)$ and set the expression equal to the remainder:

$$(-3)^2 + b(-3) + 2 = 5$$

Calculate the square and the product:

$$9 - 3b + 2 = 5$$

Combine the constant terms and isolate $b$:

$$11 - 3b = 5$$ $$-3b = 5 - 11$$ $$-3b = -6$$ $$b = \frac{-6}{-3}$$ $$b = 2$$

The value of $b$ is $\boxed{\text{2}}$.