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