The factor of the polynomial  $6x^4+29x^3+30x^2-11x-6$

$f\left(-2\right)=6{\left(-2\right)}^4+29{\left(-2\right)}^3+30{\left(-2\right)}^2-11\left(-2\right)-6$

$=96-232+120+22-6$

$=-238+238$

$=0$

$\therefore \left(x+2\right)$  is a factor of $6x^4+29x^3+30x^2-11x-6$

$f\left(\frac{1}{2}\right)=6\times \frac{1}{16}+29\times \frac{1}{8}+30\times \frac{1}{4}-11\times \frac{1}{2}-6$

$=\frac{6}{16}+\frac{29}{8}+\frac{30}{4}-\frac{11}{2}-6$

$=\frac{6+58+120-88-96}{16}$

$=\frac{184-184}{16}$

$=0$

$\therefore \left(2x-1\right)$  is a factor of $6x^4+29x^3+30x^2-11x-6$

$f\left(2\right)=6{\left(2\right)}^4+29{\left(2\right)}^3+30{\left(2\right)}^2-11\left(2\right)-6$

$=96+232+120-22-6$

$=420$

$\left(x-2\right)$  is not a factor of $6x^4+29x^3+30x^2-11x-6$

$f\left(-3\right)=6{\left(-3\right)}^4+29{\left(-3\right)}^3+30\left(-3\right)-11\left(-3\right)-6$

$=486-783-90+33-6$

$=519-873$

$=-354$

$\left(x+3\right)$  is not a factor of $6x^4+29x^3+30x^2-11x-6$