The number of distinct real roots of the equation 3 x ^ 4 + 4 x ^ 3...

$3{x}^{4} + 4{x}^{3} - 12{x}^{2} + 4 = 0$

So,

let $f\left(x\right) = 3{x}^{4} + 4{x}^{3} - 12{x}^{2} + 4$

$∴ {f}^{'}\left(x\right) = 12x\left({x}^{2} + x - 2\right)$

$= 12x\left(x + 2\right)\left(x - 1\right)$

$f\left(0\right) = 4$

$f\left(1\right) = - 1$

Ans. $4$