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This is similar to FOIL -- it's a matching game.  We just have more guys to match.

Check it out:

( 2x + 3 ) ( 4x^2 - x + 5 )

We'll start by matching the 2x to each guy in the second chunk:

( 2x + 3 ) ( 4x^2 - x + 5 ) = 8x^3 - 2x^2 + 10x ...

Then, we match the 3 to each guy in the second in the second chunk:

( 2x + 3 ) ( 4x^2 - x + 5 ) = 8x^3 - 2x^2 + 10x + 12x^2 - 3x + 15

Now, just add like terms...

= 8x^3 + 10x^2 + 7x + 15


TRY IT:

Multiply

( x + 3 ) ( 4x^2 - 2x + 6 )