TITLE: How would I solve this higher degree equation?
QUESTION [5 upvotes]: $$(x^{ 2 }-2x+3)(x^{ 2 }-2x+3)=121$$
Steps I took:
$$x^4-2x^3+3x^2-2x^3+4x^2-6x+3x^2-6x+9=121$$
$$x^{ 4 }-4x^{ 3 }+10x^{ 2 }-12x+9=121$$
$$x^{ 4 }-4x^{ 3 }+10x^{ 2 }-12x-112=0$$
At this point I don't see how I can possibly find the solution using either the quadratic formula, factoring, or completing the square. Please do not provide the solution. I only want hints to guide me in the right direction.

REPLY [2 votes]: Note that $(x^2 - 2x + 3)(x^2 - 2x + 3) = (x^2 - 2x + 3)^2$. Writing it this way will allow you to square-root both sides and then apply your other procedures.
As a lesson, don't just blindly simplify everything as you tried to do; be on the lookout for opportunities to write things as squares (frequently binomial squares), which is really the core idea behind the method of square roots and completing the square.