When product of 2 numbers is positive/negative

The sign of a product of two factors depends only on the signs of those factors: same signs give a positive product, opposite signs give a negative product.

So "the product is negative" means the two factors have opposite signs, and you check both cases one at a time.
For $(x + 1)(x - 4) < 0$, either $x + 1 > 0$ and $x - 4 < 0$, giving $x > -1$ and $x < 4$, so $-1 < x < 4$; or $x + 1 < 0$ and $x - 4 > 0$, giving $x < -1$ and $x > 4$, which is impossible.
So the product is negative for $-1 < x < 4$.

Likewise, "the product is positive" means the two factors share a sign, and again you check both cases.
For $(x + 2)(x - 3) > 0$, either both are positive, $x > -2$ and $x > 3$, giving $x > 3$; or both are negative, $x < -2$ and $x < 3$, giving $x < -2$.
So the product is positive for $x < -2$ or $x > 3$.

So the recipe is always the same: decide whether you need opposite signs (for a negative product) or the same sign (for a positive product), then work through each case and combine the results.
Take it one factor at a time and keep track of which values of $x$ satisfy both conditions in a case.
There is no shortcut — the sign rule plus careful case-checking is the whole method.

For which values of $x$ is $(x - 2)(x + 3) < 0$?
A product is negative only when the two factors have opposite signs, so check both cases.
Case 1: $x - 2 > 0$ and $x + 3 < 0$ means $x > 2$ and $x < -3$, which is impossible.
Case 2: $x - 2 < 0$ and $x + 3 > 0$ means $x < 2$ and $x > -3$, so $-3 < x < 2$, and that is the answer.

For which values of $x$ is $(x - 1)(x - 5) > 0$?
A product is positive only when the two factors have the same sign, so check both cases.
Case 1: both positive, $x - 1 > 0$ and $x - 5 > 0$, means $x > 1$ and $x > 5$, so $x > 5$.
Case 2: both negative, $x - 1 < 0$ and $x - 5 < 0$, means $x < 1$ and $x < 5$, so $x < 1$; combining, the answer is $x < 1$ or $x > 5$.

If $mn < 0$ and $m > 0$, what is the sign of $n$?
Since $mn < 0$, the product is negative, so $m$ and $n$ must have opposite signs.
We are told $m$ is positive, so $n$ must be the opposite: negative.
Therefore $n < 0$.