$$x^4+ 6x^3+13x^2+13x-1=\textrm{perfect square}$$ I don't know how to approach this. I have tried to factor this but was unable to, and equating it to squares of numbers is a tedious process, and I am not sure how to solve this problem.

By the way, this question is from Elementary Algebra by Hall and Knight, Exercise XXX b question 15.

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    $\begingroup$ Is there a perfect square that is close to your expression? A number of the form $(x^2+ax+b)^2$ perhaps? $\endgroup$
    – player3236
    Nov 17 '20 at 16:56
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    $\begingroup$ In Magma run IntegralQuarticPoints([1, 6, 13, 13, -1]); and get two solutions. $\endgroup$ Nov 17 '20 at 17:04
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    $\begingroup$ @J.W.Tanner : The answer is always 42 :-) $\endgroup$
    – Rüdi Jehn
    Nov 17 '20 at 17:15
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    $\begingroup$ I removed the abstract-algebra tag $\endgroup$ Nov 17 '20 at 17:31
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    $\begingroup$ That book does not use modular arithmetic. In fact, the article before that problem explains how to find square roots of expressions like these. For anyone interested in posting a solution the book can be found at ia800302.us.archive.org/17/items/elementaryalgebr00hall/… $\endgroup$
    – John Douma
    Nov 17 '20 at 17:32


Can you show that $(x^2+3x+2)^2<x^4+6x^3+13x^2+13x-1<(x^2+3x+3)^2$ for $x>5$?

(A suggestion in this direction was provided by player3236 in a comment.)


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