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masters Let $\displaystyle f(x)=3x^4+4x^3-8x-2$. Since f is a polynomial, it is continuous on $\displaystyle [1, 2]$.
We have $\displaystyle f(1)=3(1)^4+4(1)^3-8x-2=-3 < 0 \ \ and \ \ f(2)=3(2)^4+4(2)^3-8x-2=62 > 0$, so that $\displaystyle f(1)<0<f(2)$.
So by the intermediate-value theorem there exists $\displaystyle x_1$ in $\displaystyle (1, 2)$ such that $\displaystyle f(x_1)=0$.
That is, the equation $\displaystyle 3x^4+4x^3-8x-2=0$ has a solution in the interval $\displaystyle (1, 2)$.