Mister Exam
Factor y^4+12*y^2-15 squared
The solution
Factorization
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/ ____________\ / ____________\ / _____________\ / _____________\ | / ____ | | / ____ | | / ____ | | / ____ | \x + I*\/ 6 + \/ 51 /*\x - I*\/ 6 + \/ 51 /*\x + \/ -6 + \/ 51 /*\x - \/ -6 + \/ 51 /
$$\left(x - i \sqrt{6 + \sqrt{51}}\right) \left(x + i \sqrt{6 + \sqrt{51}}\right) \left(x + \sqrt{-6 + \sqrt{51}}\right) \left(x - \sqrt{-6 + \sqrt{51}}\right)$$
(((x + i*sqrt(6 + sqrt(51)))*(x - i*sqrt(6 + sqrt(51))))*(x + sqrt(-6 + sqrt(51))))*(x - sqrt(-6 + sqrt(51)))
The perfect square
Let's highlight the perfect square of the square three-member
$$\left(y^{4} + 12 y^{2}\right) - 15$$
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
$$a = 1$$
$$b = 12$$
$$c = -15$$
Then
$$m = 6$$
$$n = -51$$
So,
$$\left(y^{2} + 6\right)^{2} - 51$$
$$\left(y^{4} + 12 y^{2}\right) - 15$$
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
$$a = 1$$
$$b = 12$$
$$c = -15$$
Then
$$m = 6$$
$$n = -51$$
So,
$$\left(y^{2} + 6\right)^{2} - 51$$
Combining rational expressions
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2 / 2\ -15 + y *\12 + y /
$$y^{2} \left(y^{2} + 12\right) - 15$$
-15 + y^2*(12 + y^2)