<s>[INST] How is $x(2x+7)+3$ equal to $(2x+1)(x+3)$? For some reason, $x(2x+7)+3$ seems like it should be equal to $(2x+7)(x+3)$ instead of $(2x+1)(x+3)$. How does $(2x+1)$ factor out of here?
The original equation was $2x^2+7x+3$.
Proof of this: https://www.desmos.com/calculator/2xpcqznkio
Notice how $x(2x+7)+3$ and $(2x+1)(x+3)$ overlap, but $(2x+7)(x+3)$ does not!
 [/INST] It is impossible for $2x^2+7x+3$ to factor into $(2x+7)(x+3)$, because when you expand the brackets, the constant term is $21$ — not $3$. 
A way to factor quadratics without the quadratic formula, where the leading coefficient $a \ne  1$ is like this:
$$
\begin{array} {c|c|c}
2x^2 & ? \\ \hline
? & 3 \\ 
\end{array}
$$
Here we have to 'split' the $x$ term, or $7x$ into the remaining two boxes. Try splitting $x$ in different ways - one might be $x$ and $6x$, like this:
$$
\begin{array} {c|c|c}
2x^2 & x \\ \hline
6x & 3 \\ 
\end{array}
$$
However, $2x^2$ factors into $x(2x+1)$, and $6x+3$ factors into $3(x+2)$, which doesn't match. Can you find a way to split the $7x$, so that the two rows have the same factor?
</s>