How to show that linear map is surjective? I have the following linear map:
$$T:\mathcal{P}^2 \to \Bbb{R}^2 $$
where $\mathcal{P}^2$ denotes the vector space of polynomials  with real coefficients having degree at most $2$
$ T $ is defined  by $$T(ax^2+bx+c)=[a+b , b−c]^t$$

I do not know how to prove that $T$ is surjective.
I know its not injective.
Yet,i do not now how to formally show that is a surjection.
I tried following the answer at How to show that a linear map is surjective?
Using the formula described:

*

*$\dim(V) = 2$,

*$\dim(\operatorname{range}T)=1$
According to the formula "will be surjective if $\dim V= \dim \operatorname{range} T$"
this map would not be surjective.
Yet, in the website i took this exercise from, it says this map is a surjective.
So, i am wondering what i did wrong, and if there are better ways to show that this map is a surjective.
I came across on this exercise on this website.

For any $(p,q)\in\mathbb R^2$, $T(px^2-q)=(p+0,0--q)=(p,q)$. Since this is true for any pair, $T$ is surjective.
$T(x^2+5x-3)=T(2x^2+4x+2)=(6,2)$ so $T$ is not injective.
