Given\:solutions:[tex]\frac{5+2\sqrt{7}}{3}\:and\:\frac{5-2\sqrt{7}}{3}[/tex].
Therefore, factors of the equation would be [tex](x-\frac{5+2\sqrt{7}}{3})\:and\:(x-\frac{5-2\sqrt{7}}{3})[/tex]
Let us multiply those two factors to get the equation, we get
[tex]\left(x-\frac{5+2\sqrt{7}}{3}\right)\left(x-\frac{5-2\sqrt{7}}{3}\right)[/tex]
[tex]\mathrm{Apply\:FOIL\:method}:\quad \left(a+b\right)\left(c+d\right)=ac+ad+bc+bd[/tex]
[tex]a=x,\:b=-\frac{5+2\sqrt{7}}{3},\:c=x,\:d=-\frac{5-2\sqrt{7}}{3}[/tex]
[tex]=xx+x\left(-\frac{5-2\sqrt{7}}{3}\right)+\left(-\frac{5+2\sqrt{7}}{3}\right)x+\left(-\frac{5+2\sqrt{7}}{3}\right)\left(-\frac{5-2\sqrt{7}}{3}\right)[/tex]
[tex]=xx-\frac{5-2\sqrt{7}}{3}x-\frac{5+2\sqrt{7}}{3}x+\frac{5+2\sqrt{7}}{3}\cdot \frac{5-2\sqrt{7}}{3}[/tex]
[tex]=x^2-\frac{5x-2\sqrt{7}x}{3}-\frac{5x+2\sqrt{7}x}{3}-\frac{1}{3}[/tex]
[tex]\mathrm{Combine\:the\:fractions\:}-\frac{5x-2\sqrt{7}x}{3}-\frac{5x+2\sqrt{7}x}{3}-\frac{1}{3}:\quad \frac{-\left(5x-2\sqrt{7}x\right)-\left(5x+2\sqrt{7}x\right)-1}{3}[/tex]
[tex]=x^2+\frac{-\left(5x-2\sqrt{7}x\right)-\left(5x+2\sqrt{7}x\right)-1}{3}[/tex]
[tex]\mathrm{Expand}\:-\left(5x-2\sqrt{7}x\right)-\left(5x+2\sqrt{7}x\right)-1:\quad -10x-1[/tex]
[tex]=x^2+\frac{-10x-1}{3}[/tex]
Setting it equal to 0.
[tex]x^2+\frac{-10x-1}{3}=0[/tex]
Multiplying whole equation by 3, we get
[tex]3x^2-10x-1 = 0[/tex]
Therefore, correct option is D [tex]3x^2-10x-1 = 0[/tex]
