Is ($(x-1)(x-2)$) a maximal ideal in $\mathbb{R}[x]$?

Question

I dont know where to start from .Some hints would be really appreciated. In fact I want to know how do I find the maximal ideals in $\mathbb{R}[x]/((x-1)(x-2)$.How do I do that?

Answer 1

HINT: Let $R$ be a commutative ring with $1$ and let $I\subset R$ be an ideal. Then there is a natural inclusion preserving bijection between the following two sets:

1. The set of ideals of the quotient $R/I$,

2. The set of the ideal $J\subset R$ such that $I\subset J$,

given by taking the preimage under the canonical quotient map $R\rightarrow R/I$.

Answer 2

Hint (for the question in the title)

Is the ideal

$$\left((x-1)(x-2), x-1\right)$$ equal to $\mathbb R[x]$?