Mister Exam
Factor -y^4+5*y^2-11 squared
The solution
Factorization
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$$\left(x + \left(\sqrt[4]{11} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{19}}{5} \right)}}{2} \right)} - \sqrt[4]{11} i \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{19}}{5} \right)}}{2} \right)}\right)\right) \left(x + \left(\sqrt[4]{11} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{19}}{5} \right)}}{2} \right)} + \sqrt[4]{11} i \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{19}}{5} \right)}}{2} \right)}\right)\right) \left(x + \left(- \sqrt[4]{11} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{19}}{5} \right)}}{2} \right)} + \sqrt[4]{11} i \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{19}}{5} \right)}}{2} \right)}\right)\right) \left(x + \left(- \sqrt[4]{11} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{19}}{5} \right)}}{2} \right)} - \sqrt[4]{11} i \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{19}}{5} \right)}}{2} \right)}\right)\right)$$
(((x + 11^(1/4)*cos(atan(sqrt(19)/5)/2) + i*11^(1/4)*sin(atan(sqrt(19)/5)/2))*(x + 11^(1/4)*cos(atan(sqrt(19)/5)/2) - i*11^(1/4)*sin(atan(sqrt(19)/5)/2)))*(x - 11^(1/4)*cos(atan(sqrt(19)/5)/2) + i*11^(1/4)*sin(atan(sqrt(19)/5)/2)))*(x - 11^(1/4)*cos(atan(sqrt(19)/5)/2) - i*11^(1/4)*sin(atan(sqrt(19)/5)/2))
The perfect square
Let's highlight the perfect square of the square three-member
$$\left(- y^{4} + 5 y^{2}\right) - 11$$
To do this, let's use the formula
$$a y^{4} + b y^{2} + c = a \left(m + y^{2}\right)^{2} + n$$
where
$$m = \frac{b}{2 a}$$
$$n = \frac{4 a c - b^{2}}{4 a}$$
In this case
$$a = -1$$
$$b = 5$$
$$c = -11$$
Then
$$m = - \frac{5}{2}$$
$$n = - \frac{19}{4}$$
So,
$$- \left(y^{2} - \frac{5}{2}\right)^{2} - \frac{19}{4}$$
$$\left(- y^{4} + 5 y^{2}\right) - 11$$
To do this, let's use the formula
$$a y^{4} + b y^{2} + c = a \left(m + y^{2}\right)^{2} + n$$
where
$$m = \frac{b}{2 a}$$
$$n = \frac{4 a c - b^{2}}{4 a}$$
In this case
$$a = -1$$
$$b = 5$$
$$c = -11$$
Then
$$m = - \frac{5}{2}$$
$$n = - \frac{19}{4}$$
So,
$$- \left(y^{2} - \frac{5}{2}\right)^{2} - \frac{19}{4}$$
Combining rational expressions
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2 / 2\ -11 + y *\5 - y /
$$y^{2} \left(5 - y^{2}\right) - 11$$
-11 + y^2*(5 - y^2)