Expresión del cuadrado perfecto
Expresemos el cuadrado perfecto del trinomio cuadrático
$$\left(x^{4} - x^{2}\right) + 4$$
Para eso usemos la fórmula
$$a x^{4} + b x^{2} + c = a \left(m + x^{2}\right)^{2} + n$$
donde
$$m = \frac{b}{2 a}$$
$$n = \frac{4 a c - b^{2}}{4 a}$$
En nuestro caso
$$a = 1$$
$$b = -1$$
$$c = 4$$
Entonces
$$m = - \frac{1}{2}$$
$$n = \frac{15}{4}$$
Pues,
$$\left(x^{2} - \frac{1}{2}\right)^{2} + \frac{15}{4}$$
/ / / ____\\ / / ____\\\ / / / ____\\ / / ____\\\ / / / ____\\ / / ____\\\ / / / ____\\ / / ____\\\
| ___ |atan\\/ 15 /| ___ |atan\\/ 15 /|| | ___ |atan\\/ 15 /| ___ |atan\\/ 15 /|| | ___ |atan\\/ 15 /| ___ |atan\\/ 15 /|| | ___ |atan\\/ 15 /| ___ |atan\\/ 15 /||
|x + \/ 2 *cos|------------| + I*\/ 2 *sin|------------||*|x + \/ 2 *cos|------------| - I*\/ 2 *sin|------------||*|x + - \/ 2 *cos|------------| + I*\/ 2 *sin|------------||*|x + - \/ 2 *cos|------------| - I*\/ 2 *sin|------------||
\ \ 2 / \ 2 // \ \ 2 / \ 2 // \ \ 2 / \ 2 // \ \ 2 / \ 2 //
$$\left(x + \left(\sqrt{2} \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{15} \right)}}{2} \right)} - \sqrt{2} i \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{15} \right)}}{2} \right)}\right)\right) \left(x + \left(\sqrt{2} \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{15} \right)}}{2} \right)} + \sqrt{2} i \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{15} \right)}}{2} \right)}\right)\right) \left(x + \left(- \sqrt{2} \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{15} \right)}}{2} \right)} + \sqrt{2} i \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{15} \right)}}{2} \right)}\right)\right) \left(x + \left(- \sqrt{2} \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{15} \right)}}{2} \right)} - \sqrt{2} i \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{15} \right)}}{2} \right)}\right)\right)$$
(((x + sqrt(2)*cos(atan(sqrt(15))/2) + i*sqrt(2)*sin(atan(sqrt(15))/2))*(x + sqrt(2)*cos(atan(sqrt(15))/2) - i*sqrt(2)*sin(atan(sqrt(15))/2)))*(x - sqrt(2)*cos(atan(sqrt(15))/2) + i*sqrt(2)*sin(atan(sqrt(15))/2)))*(x - sqrt(2)*cos(atan(sqrt(15))/2) - i*sqrt(2)*sin(atan(sqrt(15))/2))