Usually we don't even consider the sign of the answer of a multiplication problem.
This is probably because most real-world multiplication problems are the result of multiplying two psositive numbers.
The results we're taught to memorize from the multiplication tables are always positive.
So the answer to the first question above is:  Positive times Positive equal Positive.

What about the multiplication of two negative numbers or the multiplication of a positive and a negative number?
When it comes to explaining the outcome of these two cases, I like to use the number line and the analogy of going in a positive or negitive direction.
Allow me to explain...

First, think of a positive number (+) as staying the course or not changing direction.  So if you're going in a positive direction (the first number is positive) and you multiply by a second number (that is also positive), the second positive will not "change the direction" you are currently traveling and so you continue moving in the positive direction.  So your overall answer is positive.  The following number line is only to represent "direction" of the answer.  Positive answers move to the right and negative answers move to the left.  Here, 4 • 3 keeps moving to the right and so the answer is (positive) 12.  

Now think of a negative number (─) as making a U-turn or reversing direction.  So if you start out with a negative number, you're going in the negative direction.  If you multiply by a second negative number, you make a U-turn and now your headed in the positive direction.  So your overall answer in positive.  Look at the following nuber line where two negative numbers are being multiplied.  Here, (─2) • (─5) orignally moves in the negative direction and then makes a U-turn to move in the positive direction.

In our final example, we multiply a positive time a negative number or the reverse (a negative times a positive number).  If we start with a positive number, we are moving to the right into positive territory.  Once we multiply times the second negative number, we make a U-turn and now we're heading in the negative direction.  Here, 7 • (-1) originally move in the positive direction and then changes direction and moves to becoming negative.

Or the other way around, beginning with the negative number and multiplying by a positive number.  Here, (-6) •  9 originally moves in the negative direction.  When it is multiplied by the positive number, it keeps moving in the negative direction because positive numbers stay their course.  Only negative numbers make a U-turn.