Come up with a real inner product space $V$ and $T\in \mathcal L(V) $ and real numbers b, c with $ b^2 < 4c $ so that $T^2 + bT +cI$ is not invertible

Question

I've been self studying Linear Algerba Done Right, and I have had past exposure to linear algebra( college level), although it wasn't proof based. I am struggling with this exercise: Exercise 1 https://i.stack.imgur.com/zKJRt.png

this is the solution proposed by Axler2

https://i.stack.imgur.com/Zwlx0.png

My attempt was this:

Define a linear map $T$ with respect to the standard basis in $\Re^3$ which has the matrix :

$$ \begin{bmatrix} 1 & 2 & 0 \\ 0 & 1 & 0 \\ 1 & 2 & 1 \\ \end{bmatrix} $$

(notice the matrix is not equal to its traspose, $T^*$) then proceed to compute the result of $$T^2 +bT +cI$$( choosing $b=1=c$) , yet the result is invertible. Should I have chosen an operator which wasn't invertible from the start(i.e. non injective or surjective)?, or is it the choice of $b,c$? I thought that the operator not being self adjoint would guarantee the result together with a choice of b,c which respected the inequality.

Furthermore,what is the thought process of coming up with a rotation to solve the problem, or more generally, a map that allows us to solve this?