Solve x4 + x3 + x2 + x = 4.66.

Step-by-step solution

1. 1

   Establish how many real roots to look for. With $g(x)=x(x+1)(x^2+1)$, the factor $x^2+1$ never vanishes, so $g<0$ only on $(-1,0)$ and $g\to+\infty$ in both directions. The horizontal line $y=4.66$ therefore cuts the graph exactly twice, and any answer listing only one root is incomplete.

2. 2

   Bracket the positive root. $g(1)=4$ and $g(1.1)=5.1051$, so the root lies in $(1,1.1)$ — nearer the lower end, since $4.66$ is closer to $4$ than to $5.11$.

3. 3

   Refine the positive root.

   $x_1=1.0620313,\qquad g(x_1)=4.660000$

   The reported $1.06463$ gives $g=4.6894$, off by $0.03$; with $g'\approx 11$ near this point, that corresponds to an $x$ error of about $0.0026$.

4. 4

   Bracket the negative root. $g(-1.7)=4.6291$ (just under the target) and $g(-1.75)=5.3320$ (over it), so the root is in $(-1.75,-1.7)$.

5. 5

   Refine the negative root.

   $x_2=-1.7023038,\qquad g(x_2)=4.660000$

   A value of $-1.72880$ lies inside the bracket but gives $g=5.0256$, clearly above $4.66$, so it is not the root either.

6. 6

   Collect the answer. The two real solutions are $x\approx 1.0620313$ and $x\approx -1.7023038$; the other two roots of this quartic are complex conjugates.