The textbook states that the solution is:
Let P(x) be "x is positive" and Q(x) is "x is negative". The domain is integers.
This shows $\exists x P(x)\land\exists x Q(x)$ is True and shows $\exists x(P(x)\land Q(x) )$ is False.
I take this to mean that $\exists x P(x)\land\exists x Q(x)$ is translated into English as "There exist positive integers and there exist negative integers", which is obviously true.
I think that $\exists x(P(x)\land Q(x) )$ means "There exists an integer that is positive and negative" which is False.
Does all this mean what I think it means?
You're absolutely right, and the comments to your question are probably more interesting than my answer. Still, I'd like to make the point of scopes.
The scope of a logical connective corresponds to the statements that it connects.
For instance, in $A\lor B$, it should be clear to you that both $A$ and $B$ are in the scope of $\lor$: that's how you evaluate it. Scopes become your first priority as soon as the proposition is a bit more complex. Take $A\lor B\land C$ for example. As you know, this is not a well-formed formula of sentential logic. However, adding parentheses helps you decide whether $(A\lor B)\land C$ or $A\lor(B\land C)$ is meant: the scopes of both connectives are then well-defined.
The same rules apply to quantifiers, both existential and universal.
As you correctly pointed out, it's easy to understand through a simple example how $\exists x Px\land\exists xQx$ and $\exists x(Px\land Qx)$ differ. In the second case, the scope of the existential quantifier is extended to the inside of the brackets: both the $x$ in $Px$ and $Qx$ refer to the same variable $x$, instantiated by the existential quantifier.
On the contrary, in the first case, the scopes of the existential quantifiers are limited by the conjunction, i.e.: $$(\exists x\underbrace{Px}_{\text{scope 1}})\land(\exists x\underbrace{Qx}_{\text{scope 2}})$$ which could as well be rewritten with a new variable to avoid confusion: $$\exists xPx\land\exists yQy$$ To go even further, this is equivalent to: $$\exists x\exists y(Px\land Qy)$$ (This is referred to as Prenex Notation).