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I need to find two subrings $S$ and $T$ of the polynomial ring $\mathbb{R}[x,y]$ such that $S+T$ is not a subring of $\mathbb{R}[x,y]$.

This is the ring of polynomials with real coefficients in the two variables $x$ and $y$. In order for $S+T$ to fail to be a subring, I need to show that it either doesn't contain $0$ or it isn't closed under negation, addition, or multiplication. But I'm having trouble coming up with subrings that will work.

Even hints would be helpful provided you're willing to answer follow-up questions!

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You may consider the subrings $S=\mathbb{R}[x]$ and $T=\mathbb{R}[y]$: $S+T$ consists of polynomials of the form $P(x)+Q(y)$, and it should be easy to show that it isn't closed under multiplication (look at $xy$).

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    but then $\mathbb{R}[x,y]$ itself wouldn't be closed under multiplication and yet it's a ring. – 2017-02-24
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    No, because $\mathbb{R}[x,y]$ consists by definition of all linear combinations of $x^{i}y^{j}$, not only of the $x^{i}$ and $y^{j}$ taken separately. – 2017-02-24