Example 5 If $2 x^{2}+3 x y+2 y^{2}=1$, find the minimum value of $k=x+y+x y$.

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Example 5 If $2 x^{2}+3 x y+2 y^{2}=1$, find the minimum value of $k=x+y+x y$.
Solution: From $2 x^{2}+3 x y+2 y^{2}=1$, we get $2(x+y)^{2}=1+x y$, which means $2(x+y)^{2}=k-(x+y)+1$. Rearranging terms, we have
$$
2(x+y)^{2}+(x+y)-(1+k)=0.
$$

Considering this equation as a quadratic equation in terms of $x+y$, and since $x+y$ is a real number, we have
$$
\Delta=1+8(1+k) \geqslant 0.
$$

Solving this, we get $k \geqslant-\frac{9}{8}$.
Therefore, the minimum value of $x+y+x y$ is $-\frac{9}{8}$.