<s>[INST] Understanding Heron's formula proof 
I was trying to understand the proof of Heron's formula. I understood everything up to step $(d)$ except how they arrived at the expression ($a$+$b$+$c$)($b$+$c$-$a$)($a$+$b$-$c$)($a$-$b$+$c$) and the steps afterwards I still don't understand I would appreciate if someone gave me insight into how the expressions came about
 [/INST] Hint:
\begin{align}
h^2
&=b^2 - d^2\\
&=b^2 - \frac{(b^2+c^2-a^2)^2}{4c^2}\\
&=\frac{(2bc)^2-(b^2+c^2-a^2)^2}{4c^2}\\
&=\frac{(2bc+b^2+c^2-a^2)(2bc-b^2-c^2+a^2)}{4c^2}\\
&=\frac{\big[(b+c)^2-a^2\big]\big[-(b-c)^2+a^2\big]}{4c^2}\\
&=\frac{(a+b+c)(-a+b+c)(a-b+c)(a+b-c)}{4c^2}
\end{align}
</s>