$$\left(x+\dfrac{b}{2a}\right)^2 = \dfrac{-4ac+b^2}{4a^2}$$
goes to :
$$\left(x+\dfrac{b}{2a}\right)^2 = \dfrac{b^2-4ac}{4a^2}$$
I don't understand how the signs changed in going from -4ac+b^2 to b^2-4ac.
I am a beginner at maths ; I have tried to understand this but I can't make sense of this
Think of it as addition but one of the arguments is negative, i.e., $a + (-b) $ = $ (-b) + a$ holds for all $a, b \in \mathbb{R}$.
Since the OP asked for a proof, this may have way more details than are really important...
Although the short version is that this is just a reordering of terms, there is a little bit of subtlety here because of the subtraction. After all, $a - b$ and $b - a$ are most definitely not the same because subtraction is sensitive to order.
But here, we're ok. Recall that $y - x$ is really shorthand for $y + (-x)$, where $(-x)$ is the additive inverse of $x$ (the thing that satisfies $x + (-x) = 0$). So we have
$$y - x = y + (-x) = (-x) + y$$
because addition is commutative. Now we normally don't bother to write the parentheses here, so this is just $-x + y$ as desired.