Counting the Negatives

A long product is worked one pair at a time, left to right. $\left(-4\right) \cdot 2 \cdot \left(-1\right) \cdot \left(-3\right)$: first $\left(-4\right) \cdot 2$ is -8, giving $-8 \cdot \left(-1\right) \cdot \left(-3\right)$. Then $-8 \cdot \left(-1\right)$ is 8, giving $8 \cdot \left(-3\right)$. Then -24.

You can know the sign before doing any of that. Each negative factor flips the sign, and two flips cancel. So count the negative factors: an even number of them gives a positive product, an odd number gives a negative one. The example has three, so the answer had to be negative.

That splits the job in two: count the negatives for the sign, and multiply the sizes for the number. 4, 2, 1 and 3 multiply to 24, and three negatives make it -24. It is also a quick way to check your last line.