If [tex]3x-4[/tex] is a factor of [tex]p(x)=-6x^3-7x^2+qx-12[/tex], then [tex]x=\frac43[/tex] is a root of [tex]p(x)[/tex] so that [tex]p\left(\frac43\right)=0[/tex]:
[tex]-6\left(\dfrac43\right)^3-7\left(\dfrac43\right)^2+q\left(\dfrac43\right)-12=0[/tex]
[tex]\dfrac{4q-116}3=0[/tex]
[tex]4q-116=0[/tex]
[tex]4(q-29)=0[/tex]
[tex]\implies\boxed{q=29}[/tex]
