Is there a quick way to expand: (3x + 5)(x - 7)?
This is a good example where you may say "and" instead of "plus".
$\displaystyle (2+3)(1+5)=5(6)=30$
We may also write
$\displaystyle (2+3)(1+5)=2(1+5)$ and $\displaystyle 3(1+5)$ or $\displaystyle 1(2+3)$ and $\displaystyle 5(2+3)$
which is
$\displaystyle 2(1)$ and $\displaystyle 2(5)$ and $\displaystyle 3(1)$ and $\displaystyle 3(5)$
$\displaystyle 2+10+3+15=30$
The whole is the sum of the parts
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$\displaystyle (3x+5)(x-7)=3x(x-7)+5(x-7)=3x(x)+3x(-7)+5(x)+5(-7)$
which is the 3x multiplied by both in the 2nd brackets and 5 by both in the 2nd brackets.