Suppose we have $R=\mathbb{R}[x]$ the ring of polynomials and the ideals $I=(x^2+1)$ and $J=(x+1)$. We need to show that $I$ and $J$ are co-maximal, so we want $I + J = R$.
I have absolutely no idea how to show this. For example, how do I show that $5x^2+1 \in I + J$? We know $5x^2+5 \in I+J$ because it's an element of $I$. But I don't see how we can show that all the elements of R are in $I+J$.
You only have to show that a unit, like $1$ simply, is contained in your ideal. This can be done directly, via something like $\frac{1}{2}(x^2+1) - \frac{1}{2}(x-1)(x+1) = 1 \in I + J$. Additionally, since $\mathbb{R}$ is a field $\mathbb{R}[x]$ is a Principal Ideal Domain (and also a Unique Factorization Domain), and you use the fact that $x^2+1, x+1$ are coprime (since they are essentially distinct irreducibles).