How to convert a nonstandard expression to form $(x-b)^2$

Question

I want to convert the expression $x(−e_c^2+1)(x+2ae_c)$ into form $(x−b)^2$, where $a$ does not have any $x$ in it. I've tried converting them with completing the square, but I always get at least one term with an $x$ in it. Is this possible, and if so, how do I do it, and what should my result be?

Answer 1

$$x(−e_c^2+1)(x+2ae_c)= (−e_c^2+1)(x^2+2ae_cx)= (−e_c^2+1)((x+ae_c)^2-1) $$

Answer 2

I will assume that "convert" as used in the title and body of the Question means rewriting one (polynomial) expression as another, of the form $(x-b)^2$, so that the two expressions are equal as (polynomial) functions over the domain of real numbers.

It is not possible unless $b=e_c=0$. One way to see this is that $(x-b)^2$ has a double root at $x=b$ (and no other root). Since $x(−e_c^2+1)(x+2ae_c)$ has at least one root at $x=0$, these expressions can be equal as polynomial functions only if:

$$ b = 0 \;\text{ and }\; 2ae_c = 0 $$

so that we then get a double root at $x=b=0$ for both expressions, and no other roots of either expression. Furthermore we would need the leading coefficient to be one:

$$ −e_c^2+1 = 1 $$

so that both expressions are monic polynomials.

Taken together these conditions are satisfied exactly when $b = e_c = 0$ (and it doesn't matter what $a$ is since it is multiplied by zero).