With utmost respect for Dr. Peterson, whether infinity is a "number" depends on how you define "number. Mathematicians as distinguished as Hilbert and Cantor would not agree with Dr. Peterson's definition: mathematicians like Hilbert were happy to contemplate transfinite numbers. Other mathematicians, called finitists, do agree with Dr. Peterson: infinity is not a number.
One way to define infinity as a number is through the one-point compactification of the real numbers, which is an extension of the real numbers just as real numbers are an extension of the rational numbers, the rational numbers are an extension of the integers, and the integers are an extension of the natural numbers. In the case of most extensions of the definition of number, you must learn a new arithmetic to make the extension useful and consistent. In the case of the one-point compactification, you have to have rules that specify the arithmetic applicable to infinity. See
en.wikipedia.org
Under that particular set of rules, your squaring of negative infinity is not a defined operation. Even if we define infinity in a sensible way, you must have erred in some fashion.
Now where I agree completely with Dr. Peterson is this.
Infinity is not a real number. Moreover, you do not need a compactification of the real numbers to solve any problem that I have ever encountered. The way we normally teach mathematics formally never treats infinity as a number. Here the only reason that you got embroiled in squaring negative infinity is that you introduced an unnecessary complication.
[math]log_3(x) - 5\sqrt{log_3(x)} + 3 >0 \text { and } t = \sqrt{log_3(x)} \implies t^2 - 5t + 3 > 0 = t^2 - 5|t| + 3.[/math]
I suspect that you were thinking along the lines that the square root function is never negative. But all of part II is simply unnecessary nonsense. t is necessarily non-negative. So part II is dealing with an impossibility that you introduced by inserting absolute value notation into your work. Not an error exactly, but an an unnecessary complication. You came up with negative infinity because you were preceding from a false hypothesis, namely that t was negative.