ANSWER
The correct answer is [tex]x=15\:or\:x=2[/tex]
EXPLANATION
We have the equation;
[tex]\frac{12}{x-6}+x=3+ \frac{8x}{x-6}[/tex]
We multiply through by the least common multiple, which is [tex](x-6)[/tex].
This gives us;
[tex](x-6) \times \frac{12}{x-6}+x(x-6)=3(x-6)+ \frac{8x}{x-6} \times (x-6)[/tex]
We simplify to obtain;
[tex]12+x(x-6)=3(x-6)+ 8x[/tex]
We now expand to obtain;
[tex]12+x^2-6x=3x-18+ 8x[/tex]
We rewrite the above equation as a quadratic equation in [tex]x[/tex].
This implies that
[tex]x^2-6x-3x-8x+12+18=0[/tex]
This simplifies to;
[tex]x^2-17x+30=0[/tex]
We now split the middle term to obtain;
[tex]x^2-15x-2x+30=0[/tex]
We factor to obtain;
[tex]x(x-15)-2(x-15)=0[/tex]
[tex](x-15)(x-2)=0[/tex]
[tex](x-15)=0\:or\:(x-2)=0[/tex]
[tex]x=15\:or\:x=2[/tex]
Therefore the correct answer is option C
