Multiplying with Negative Numbers

When your hand is then first swap around the holders to get which will give the same result.

We know that is flipped to give
and so gives the same results as

Recording is (-4) x 5 x 6 = (-120) = -120

Generally flipping a holder anywhere in a chain of holders will flip the card held at the top.
Two flips will return the card to its original position.

So gives the same result as

Recording (-6) x (-3) x 7 = 126

New Rule
When there are negative holders look to see if they can be paired.
When all negative holders can be put into groups of two leave the card alone.
When there is one negative holder that cannot be put in a group of two flip the card.
In both cases flip all negative holders.

Examples

Both these chains of holders have negative holders that can be grouped into twos.

 In these cases just flip all negative holders

Recording (-7) x 6 x 9 x (-2) = 252 and
(-8) x (-3) x (-5) x (-4) x (-5) = (-2400) = -2400

Both these chains of holders have a negative holder that cannot be paired with another.

In these cases flip the cards as well as all negative holders

Recording (-9) x 7 x 3 x 2 = (-278) = -278 and
(-3) x (-7) x 4 x (-6) x (-8) = 4032