Answer
$527\;\rm Hz$
Work Step by Step
We know that the beat frequency is given by
$$f_{\rm Beat}=|f_1-f_2|$$
Thus, plugging the known
$$4=|f_1-523|$$
Thus, whether $f_1\gt 523$ and hence $f_1=527$ Hz, or $f_1\lt 523$, and hence $f_1=519$ Hz.
The player is increasing the length of the flute which means increasing its standing wavelength. And since the wavelength increases, the frequency decreases. Thus her initial frequency is greater than the new frequency, $f_1\gt 523$.
Therefore,
$$f_1=523+4=\color{red}{\bf 527}\;\rm Hz$$