What multiplies to give you negative 35 but adds to give you 8

Let's just ignore the negative part first, and answer the question 'Which two numbers multiply to become 35' since you can answer this question then apply the negative sign to one of the numbers.

Let's list all the possibilities:

1 35

5 7

Now let's list the possibilites with the negative sign:

-1 35

1 - 35

-5 7

5 - 7

now try adding the pairs up. Do any of them come up as 8?

Nope.

Unless there was an error, the answers are not integers