"And, of course, as we all know, with a computer you can take a simple problem like solve 5x2+2x+1=7 but you can make it harder, for example solve 5x4+2x+1=7. The principle of the problem is still the same."
This is not, in fact, true for polynomial equations of order greater than four. From Wikipedia, "the Abel–Ruffini theorem states that there is no general algebraic solution... to polynomial equations of degree five or higher." This is often featured as one of the main results in an undergraduate abstract algebra course.
I'm all for automating calculations, but the algorithms behind calculations and their proofs are important and often fascinating. Handing students an implementation of such algorithms, e.g. Mathematica, takes away their opportunity to code up the algorithms and understand them deeply.
For that matter, asking students at the high school level to derive the quadratic formula will likely be quite challenging for even the best students in the class.
Asking students to come up with the formulas for 3rd or 4th degree polynomials on their own is probably going to be beyond anyone below the very top tier of human capability. A teacher would be extremely lucky to see one or two students that can do this in their career.
Then again, most people could easily solve these problems if they know what terms to Google :)
Anyway, as I recall the third- and fourth-degree cases require clever substitutions (whereas the derivation of QF is just completing the square with general coefficients).
1. Yes. Bouldering is one type of rock climbing.
2. Very occasionally. It does limit flexibility somewhat, but I wouldn't skip climbing because I forgot my shorts.
3. Both.
4. They're great. Just make sure to get outside eventually!
(edit: formatting)