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If [math]\alpha[/math] and [math]\beta[/math] are roots of equation [math]x^2-2px+2p^4[/math], then what is the maximum value of [math]\alpha^2+\beta^2[/math]?
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From the quadratic equation [math]x^2-2px+2p^4=0[/math], we get [math]\alpha+\beta=2p[/math] and [math]\alpha\beta=2p^4[/math]. 
Now, we need to find [math]\alpha^2+\beta^2[/math] which can be written as [math](\alpha+\beta)^2-2\alpha\beta[/math]. 
Substituting the values we got earlier, we get [math]\alpha^2+\beta^2=(2p)^2-2(2p^4)=4p^2-4p^4[/math].