Step 1: Understand the Factor Theorem. The Factor Theorem states that if $(x-c)$ is a factor of a polynomial $P(x)$, then $P(c) = 0$. Step 2: Identify the polynomial and the factor. The given polynomial is $P(x) = 2x^3+4x^2+px+7$. The given factor is $(x+2)$. Comparing $(x+2)$ with $(x-c)$, we find that $c = -2$. Step 3: Apply the Factor Theorem. Since $(x+2)$ is a factor of $P(x)$, we must have $P(-2) = 0$. Substitute $x = -2$ into the polynomial: $$P(-2) = 2(-2)^3 + 4(-2)^2 + p(-2) + 7$$ Step 4: Simplify the expression and solve for $p$. $$2(-8) + 4(4) - 2p + 7 = 0$$ $$-16 + 16 - 2p + 7 = 0$$ $$0 - 2p + 7 = 0$$ $$-2p + 7 = 0$$ $$-2p = -7$$ $$p = \frac{-7}{-2}$$ $$p = \frac{7}{2}$$ The value of $p$ is $\boxed{\frac{7}{2}}$.