If $\mathrm{x}=\frac{4}{9}$ satisfies the equation ${\log }_{\mathrm{a}}\left({\mathrm{x}}^2-\mathrm{x}+2\right)>{\log }_{\mathrm{a}}\left(-{\mathrm{x}}^2+2\mathrm{x}+3)\right.$, then the sum of all possible distinct values of [x] is (where [.] represents the greatest integer function) _____.

${\log }_{\mathrm{a}}\left({\mathrm{x}}^2-\mathrm{x}+2\right)>{\log }_{\mathrm{a}}\left(-{\mathrm{x}}^2+2\mathrm{x}+3\right)$

$\begin{array}{l}\text{ Putting }\mathrm{x}=\frac{4}{9},{\log }_{\mathrm{a}}\left(\frac{142}{81}\right)>{\log }_{\mathrm{a}}\left(\frac{299}{81}\right)\\ \because \frac{142}{81}<\frac{299}{81}\text{ or }0<\mathrm{a}<1\\ {\log }_{\mathrm{a}}\left({\mathrm{x}}^2-\mathrm{x}+2\right)>{\log }_2\left(-{\mathrm{x}}^2+2\mathrm{x}+3\right)\end{array}$

$\text{ gives }0<{\mathrm{x}}^2-\mathrm{x}+2<-{\mathrm{x}}^2+2\mathrm{x}+3$

$\text{ or }{\mathrm{x}}^2-\mathrm{x}+2>0\text{ and }2{\mathrm{x}}^2-3\mathrm{x}-1<0$

$\text{ or }\frac{3-\sqrt{17}}{4}<\mathrm{x}<\frac{3+\sqrt{17}}{4}$