Mister Exam
Factor -x^4+2*x^2+8 squared
The solution
Factorization
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/ ___\ / ___\ (x + 2)*(x - 2)*\x + I*\/ 2 /*\x - I*\/ 2 /
$$\left(x - 2\right) \left(x + 2\right) \left(x + \sqrt{2} i\right) \left(x - \sqrt{2} i\right)$$
(((x + 2)*(x - 2))*(x + i*sqrt(2)))*(x - i*sqrt(2))
The perfect square
Let's highlight the perfect square of the square three-member
$$\left(- x^{4} + 2 x^{2}\right) + 8$$
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
$$a = -1$$
$$b = 2$$
$$c = 8$$
Then
$$m = -1$$
$$n = 9$$
So,
$$9 - \left(x^{2} - 1\right)^{2}$$
$$\left(- x^{4} + 2 x^{2}\right) + 8$$
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
$$a = -1$$
$$b = 2$$
$$c = 8$$
Then
$$m = -1$$
$$n = 9$$
So,
$$9 - \left(x^{2} - 1\right)^{2}$$
Combinatorics
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/ 2\ -(-2 + x)*(2 + x)*\2 + x /
$$- \left(x - 2\right) \left(x + 2\right) \left(x^{2} + 2\right)$$
-(-2 + x)*(2 + x)*(2 + x^2)
Combining rational expressions
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2 / 2\ 8 + x *\2 - x /
$$x^{2} \left(2 - x^{2}\right) + 8$$
8 + x^2*(2 - x^2)