The product of some negative number and 10 less than three times that number is 8

So,

"The product of some negative number"
n *

"and 10 less than three times that number"
-10 + 3n or 3n - 10

"is 8"
= 8
n * (3n - 10) = 8
n(3n - 10) = 8

Distribute.


Subtract 8 from both sides.


Factor.


Set each factor to zero.
3n + 2 = 0
n - 4 = 0

There are two apparent solutions, but we must remember that the problem says that the number is negative.  We must therefore discard the second equation as extraneous.

3n + 2 = 0

Subtract 2 from both sides.
3n = -2

Divide both sides by 3.