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 number of integral values of x is

$\begin{array}{l}{\log }_{\mathrm{a}}\left({\mathrm{x}}^2-\mathrm{x}+2\right)>{\log }_{\mathrm{a}}\left(-{\mathrm{x}}^2+2\mathrm{x}+3\right)\\ \mathrm{Putting} \mathrm{x}=\frac{4}{9}, {\log }_{\mathrm{a}}\left(\frac{142}{81}\right)>{\log }_{\mathrm{a}}\left(\frac{299}{81}\right)\\ \frac{142}{81}<\frac{299}{81}, \mathrm{we} \mathrm{have} 0<\mathrm{a}<1\\ \therefore {\log }_{\mathrm{a}}\left({\mathrm{x}}^2-\mathrm{x}+2\right)>{\log }_{\mathrm{a}}\left(-{\mathrm{x}}^2+2\mathrm{x}+3\right)\\ \mathrm{gives} 0<{\mathrm{x}}^2-\mathrm{x}+2<-{\mathrm{x}}^2+2\mathrm{x}+3\\ \mathrm{or} {\mathrm{x}}^2-\mathrm{x}+2>0 \mathrm{and} 2{\mathrm{x}}^2-3\mathrm{x}-1<0\\ \mathrm{or} \frac{3-\sqrt{17}}{4}<\mathrm{x}<\frac{3+\sqrt{17}}{4}\end{array}$