Given the equation:

$mx^3 + mx^2 + mx + c = 0$

and knowing that $x = c$ is a root of the equation, we want to determine the possible values of $m$, given that $m$ and $c$ are both real numbers and neither is zero.

Step 1: Substitute $x = c$ into the equation

Since $x = c$ is a root, it satisfies the equation. Substitute $x = c$:

$m(c^3) + m(c^2) + m(c) + c = 0$

Simplify this:

$mc^3 + mc^2 + mc + c = 0$

Factor $c$ from the terms on the left side:

$c(mc^2 + mc + m + 1) = 0$

Step 2: Analyze the equation

Since $c \neq 0$, divide both sides by $c$:

$mc^2 + mc + m + 1 = 0$

Step 3: Factorize and simplify

We can rewrite the equation by factoring out $m$:

$m(c^2 + c + 1) + 1 = 0$

Solving for $m$:

$m(c^2 + c + 1) = -1$

$m = \frac{-1}{c^2 + c + 1}$

Step 4: Determine the restrictions on $m$

The denominator $c^2 + c + 1$ must not be zero since $m$ is finite. Check if $c^2 + c + 1 = 0$ has real roots:

$c = \frac{-1 \pm \sqrt{1 - 4}}{2} = \frac{-1 \pm \sqrt{-3}}{2}$

The roots are complex, so $c^2 + c + 1$ is never zero for real $c$. Therefore, the expression $m = \frac{-1}{c^2 + c + 1}$ is always defined for real $c$.

Conclusion

The possible values of $m$ are:

$m = \frac{-1}{c^2 + c + 1}, \quad \text{where } c \text{ is any real number except zero}.$

Would you like further details or have any questions?

Related Questions:

1. How does the value of $m$ change if $c$ is positive or negative?
2. What are the implications of $m$ being negative given the above formula?
3. How would the value of $m$ be affected if the root were $x = -c$ instead of $x = c$?
4. Can this equation have any repeated roots other than $x = c$?
5. How can the discriminant of a cubic equation help in analyzing the nature of its roots?

Tip: Always check if the denominator in a fraction involving variables can ever be zero to avoid undefined values in the solution.