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Math·Advanced Math·Nonlinear equations in one variable and systems of equations in two variables

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The function $h$ is defined by $h(x)=-\sqrt{x^2+bx+c}$, where $b$ and $c$ are constants. In the $xy$-plane, the graph of $y=h(x)$ contains the points $(2,0)$ and $\left(0,-\sqrt{334}\right)$. If $h(m)=0$, what is the greatest possible value of $m$?

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