 Solution

The conditions set the constant and linear coefficients to zero, so $\boxed{J_3=(X^2)}$. This is a principal ideal, but it is not maximal: there is a strict chain $(X^2)⊊(X)⊊\mathbb{R}[X]$. In the quotient ring, the nonzero class of $X$ has square zero, so the quotient is not a field.