Maximal Ideals in polynomial quotient rings

Hints:

1) You can think of the quotient ring $A = \Bbb{R}[x]/(x^2)$ as being like the polynomial ring $\Bbb{R}[x]$ but equipped with a new algebraic law saying that $x^2 = 0$. So every element of $A$ can be written uniquely in the form $a + bx$ for $a, b \in \Bbb{R}$ (and where I'm using $x$ by abuse of notation for the coset $x + (x^2)$). You add in $A$ just as for polynomials and you multiply using the rule $(a + bx)(c + dx) = ac + (ad + bc)x$.

2) An ideal $M$ in a ring $R$ is maximal iff the quotient ring $K = R/M$ is a field. In a field, $t^2 = 0$ implies $t = 0$, so an element like $x \in A$ with $x^2 = 0$ must map to $0$ in $K$, i.e., it must belong to $M$. This leaves you with only one possibility for $M$ in the ring $A$.