I am currently learning how to factorise quadratic expressions of the form $ax^2 + bx + c$.
In my textbook (Foundations math seventh edition) they are elaborating on a particular way of finding a factorised expression of a quadratic expression where $a > 1$.
It goes as follows: (1) $S = 2x^2 + 5x - 3$ ; (2) $2S = 2(2x^2) + 2(5x) - 6$ ; (3) $2S = (2x)^2 + 5 (2x) - 6$ ; after which they take $2x$ to be $z$ so that we can move further in the process with the expression $z^2 + 5z - 6$.
The steps after (3) I understand very well. But I cannot wrap my head around the way they say that $2(2x^2) + 2(5x)$ may be written as $(2x)^2 + 5(2x)$
Can someone explain why this rewrite rule can be applied: i.e. which rules allow such rewriting?
Going off my comment: We can treat it as an application of the associative and commutative properties of multiplication. We have
\begin{align} 2(2x^2) & = 2 \cdot 2 \cdot x \cdot x \\ & = 2 \cdot x \cdot 2 \cdot x \\ & = (2x)^2 \end{align}
Similarly, we have
\begin{align} 2(5x) & = 2 \cdot 5 \cdot x \\ & = 5 \cdot 2 \cdot x \\ & = 5(2x) \end{align}