SOLUTION 11: We are given the equation
$$\displaystyle{ x^4 \over x+1 } = x^2+5 \ \ \ \ \longrightarrow \ \ \ \ x^4=(x+1)(x^2+5)$$ $$\longrightarrow \ \ \ \ x^4= x^3+x^2+5x+1$$ $$\longrightarrow \ \ \ \ x^4- x^3-x^2-5x-1 =0$$
Let function
$$f(x)= x^4-x^3-x^2-5x-1 \ \ \ \ and \ choose \ \ \ \ m=0$$
This function is continuous for all values of $x$ since it is a polynomial. We now need to search for an appropriate interval satisfying the assumptions of the Intermediate Value Theorem. By trial and error, we have that
$$f(0)= (0)^4-(0)^3-(0)^2-5(0)-1 = -1 < 0 \ \ \ \ and \ \ \ \ f(3)= (3)^4-(3)^3-(3)^2-5(3)-1 =29 > 0$$
so that $$f(0) = -1 < m < 29 = f(3)$$
i.e., $m=0$ is between $f(0)$ and $f(3)$.

The assumptions of the Intermediate Value Theorem have now been met, so we can conclude that there is some number $c$ in the interval $[0, 3]$ which satisfies $$f(c)=m$$ i.e., $$c^4-c^3-c^2-5c-1 =0$$ and the equation is solvable.