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.

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