# Find the values of $x$ which will make each of the following expression a perfect square: $x^4+ 6x^3+13x^2+13x-1$.

$$x^4+ 6x^3+13x^2+13x-1=\textrm{perfect square}$$ 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.

• Is there a perfect square that is close to your expression? A number of the form $(x^2+ax+b)^2$ perhaps? Nov 17 '20 at 16:56
• In Magma run IntegralQuarticPoints([1, 6, 13, 13, -1]); and get two solutions. Nov 17 '20 at 17:04
• @J.W.Tanner : The answer is always 42 :-) Nov 17 '20 at 17:15
• I removed the abstract-algebra tag Nov 17 '20 at 17:31
• 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/… Nov 17 '20 at 17:32

Can you show that $$(x^2+3x+2)^2 for $$x>5$$?