Expresión del cuadrado perfecto
Expresemos el cuadrado perfecto del trinomio cuadrático
$$\left(y^{4} + 3 y^{2}\right) + 3$$
Para eso usemos la fórmula
$$a y^{4} + b y^{2} + c = a \left(m + y^{2}\right)^{2} + n$$
donde
$$m = \frac{b}{2 a}$$
$$n = \frac{4 a c - b^{2}}{4 a}$$
En nuestro caso
$$a = 1$$
$$b = 3$$
$$c = 3$$
Entonces
$$m = \frac{3}{2}$$
$$n = \frac{3}{4}$$
Pues,
$$\left(y^{2} + \frac{3}{2}\right)^{2} + \frac{3}{4}$$
/ ___ 4 ___ ___ 3/4 ___ 4 ___ ___ 3/4\ / ___ 4 ___ ___ 3/4 ___ 4 ___ ___ 3/4\ / ___ 3/4 ___ 4 ___ ___ 4 ___ ___ 3/4\ / ___ 3/4 ___ 4 ___ ___ 4 ___ ___ 3/4\
| \/ 2 *\/ 3 \/ 2 *3 I*\/ 2 *\/ 3 I*\/ 2 *3 | | \/ 2 *\/ 3 \/ 2 *3 I*\/ 2 *\/ 3 I*\/ 2 *3 | | \/ 2 *3 \/ 2 *\/ 3 I*\/ 2 *\/ 3 I*\/ 2 *3 | | \/ 2 *3 \/ 2 *\/ 3 I*\/ 2 *\/ 3 I*\/ 2 *3 |
|x + ----------- - ---------- - ------------- - ------------|*|x + ----------- - ---------- + ------------- + ------------|*|x + ---------- - ----------- - ------------- - ------------|*|x + ---------- - ----------- + ------------- + ------------|
\ 4 4 4 4 / \ 4 4 4 4 / \ 4 4 4 4 / \ 4 4 4 4 /
$$\left(x + \left(- \frac{\sqrt{2} \cdot 3^{\frac{3}{4}}}{4} + \frac{\sqrt{2} \sqrt[4]{3}}{4} + \frac{\sqrt{2} \sqrt[4]{3} i}{4} + \frac{\sqrt{2} \cdot 3^{\frac{3}{4}} i}{4}\right)\right) \left(x + \left(- \frac{\sqrt{2} \cdot 3^{\frac{3}{4}}}{4} + \frac{\sqrt{2} \sqrt[4]{3}}{4} - \frac{\sqrt{2} \cdot 3^{\frac{3}{4}} i}{4} - \frac{\sqrt{2} \sqrt[4]{3} i}{4}\right)\right) \left(x + \left(- \frac{\sqrt{2} \sqrt[4]{3}}{4} + \frac{\sqrt{2} \cdot 3^{\frac{3}{4}}}{4} - \frac{\sqrt{2} \cdot 3^{\frac{3}{4}} i}{4} - \frac{\sqrt{2} \sqrt[4]{3} i}{4}\right)\right) \left(x + \left(- \frac{\sqrt{2} \sqrt[4]{3}}{4} + \frac{\sqrt{2} \cdot 3^{\frac{3}{4}}}{4} + \frac{\sqrt{2} \sqrt[4]{3} i}{4} + \frac{\sqrt{2} \cdot 3^{\frac{3}{4}} i}{4}\right)\right)$$
(((x + sqrt(2)*3^(1/4)/4 - sqrt(2)*3^(3/4)/4 - i*sqrt(2)*3^(1/4)/4 - i*sqrt(2)*3^(3/4)/4)*(x + sqrt(2)*3^(1/4)/4 - sqrt(2)*3^(3/4)/4 + i*sqrt(2)*3^(1/4)/4 + i*sqrt(2)*3^(3/4)/4))*(x + sqrt(2)*3^(3/4)/4 - sqrt(2)*3^(1/4)/4 - i*sqrt(2)*3^(1/4)/4 - i*sqrt(2)*3^(3/4)/4))*(x + sqrt(2)*3^(3/4)/4 - sqrt(2)*3^(1/4)/4 + i*sqrt(2)*3^(1/4)/4 + i*sqrt(2)*3^(3/4)/4)