Expresión del cuadrado perfecto
Expresemos el cuadrado perfecto del trinomio cuadrático
$$\left(- x^{4} + x^{2}\right) + 5$$
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
$$a = -1$$
$$b = 1$$
$$c = 5$$
Entonces
$$m = - \frac{1}{2}$$
$$n = \frac{21}{4}$$
Pues,
$$\frac{21}{4} - \left(x^{2} - \frac{1}{2}\right)^{2}$$
/ ______________\ / ______________\ / ____________\ / ____________\
| / ____ | | / ____ | | / ____ | | / ____ |
| / 1 \/ 21 | | / 1 \/ 21 | | / 1 \/ 21 | | / 1 \/ 21 |
|x + I* / - - + ------ |*|x - I* / - - + ------ |*|x + / - + ------ |*|x - / - + ------ |
\ \/ 2 2 / \ \/ 2 2 / \ \/ 2 2 / \ \/ 2 2 /
$$\left(x - i \sqrt{- \frac{1}{2} + \frac{\sqrt{21}}{2}}\right) \left(x + i \sqrt{- \frac{1}{2} + \frac{\sqrt{21}}{2}}\right) \left(x + \sqrt{\frac{1}{2} + \frac{\sqrt{21}}{2}}\right) \left(x - \sqrt{\frac{1}{2} + \frac{\sqrt{21}}{2}}\right)$$
(((x + i*sqrt(-1/2 + sqrt(21)/2))*(x - i*sqrt(-1/2 + sqrt(21)/2)))*(x + sqrt(1/2 + sqrt(21)/2)))*(x - sqrt(1/2 + sqrt(21)/2))