I want to solve this problem, but I have no idea how I can start:
If $K$ is a field, $(a_1,...,a_n) \in K^n,$ and $I$ the ideal $I=\langle x_1-a_1,...,x_n-a_n\rangle$, then how can we prove that $I$ is a maximal ideal?
One example: Is $\langle x^2+1 \rangle$ a maximal ideal of $ \mathbb{R}[x]$?
For your first question, suppose that $J\supset I$ is a larger ideal. You want to show that in fact $J=K[x_1,\ldots,x_n]$. Now let $f\in J\setminus I$. Apply the division algorithm to $f$ with respect to $x_1-a_1,\ldots,x-a_n$ and let $r$ be the remainder. Why must $r\in K$? Why must $r\neq 0$? So $J$ contains a nonzero element of $K$. Why then must $J= K[x_1,\ldots,x_n]$?
For your second question, consider $\mathbb{R}[x]/\langle x^2+1\rangle.$ Can you recognize this as a certain field?