Let M be a maximal ideal of a commutative ring R with identity. Prove that for each $\displaystyle a \in R, MR[X]+(X+a)R[X] $ is a meximal ideal of R.

Proof so far.

Suppose that $\displaystyle MR[X]+(X+a)R[X] \subseteq J \subseteq R $, and I want to show that J = R.

Or should I process another way, prove that J = $\displaystyle MR[X]+(X+a)R[X] \subseteq J \subseteq R $?