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!

Let $7 = 1 + 6$, then $$\begin{align} x(2x+7) + 3 &= x(2x + 1 + 6) + 3 \\ &= x(2x + 1) + 6x + 3 \\ &= x(2x + 1) + 3(2x + 1) \\ &= (2x+1)(x+3)\end{align}$$ as should be desired.
