I'll try to make this answer a little more general than just telling how many points I would give for this particular error (if interested: I'd give 5/10 at most, most likely less).
For that, let's discuss three different kinds of computation errors (i.e. not including logical errors, wrong proofs, etc.). The given error in your image falls in the third category and thus should lose lots of points...
- Careless mistake, e.g. due to missing concentration.
This is the most common mistake, the one you might have thought about when asking this question. The student simply put a wrong sign/a wrong digit/..., but did all the right computations. Here, the general rule is: Give almost full points, unless the problem got a lot easier through the mistake.
As an example, consider the task to differentiate
$$\frac{5x^4 + 7x^2-3x}{x+3}.$$
If a student forgets to copy the three from the question sheet and instead differentiates
$$\frac{5x^4 + 7x^2 - 3x}{x} = 5x^3 + 7x^2 - 3,$$
this is a lot easier and can only give few points, even if it was an honest mistake of just forgetting to copy this one digit.
- Results that can be verified.
There are many examples of exercises where the results can be verified quite simply, e.g. computing zeros of a polynomial, computing the kernel of a matrix, etc. If there is a small error here that goes unnoticed, it should give more points off. On the other hand, if a student writes something along the line of "I put this value into the polynomial and it turns out to not be a root, however, I can't find where my computation went wrong", then doing this test should be rewarded by only taking off points according to case 1. This is also important in other classes (physics, etc.), if you for example make an error in computing the speed of an object, it should at least give partial points if you notice that this car, driving at about twice the speed of light, is highly unlikely. :)
- Mistakes due to not understanding the topic.
This is the gravest situation, and the situation you have in your image. Thus, it should give the most points off. Let's take your image as an example of what I mean. The question to write this as a single polynomial asks the student to perform two steps:
- Use the distributive law to eliminate parentheses.
- Use the distributive law to collect the powers of $x$ together.
Note that collecting the powers of $x$ can be done without understanding the distributive law.
Now this student did not understand this law and failed at the first step. It might look like only two wrong signs, but looking more closely, the student simply dropped the parentheses without doing anything else. Thus, this error is just as bad as saying
$$5(x^2+3x+7) = 5x^2+3x+7.$$
As the distributive law seems to be the current topic in class, there should have been quite a few examples discussed before the test and this student simply didn't understand this topic. Thus, you should only give few points here.
Note that it is sometimes difficult to distinguish between the first case and the third one, especially if only looking at a single question, e.g. you might consider it case one instead of three if the student did all other problems with distributive law correctly... Furthermore, it also depends on the current topic taught in class: if your given mistake happens in a graduate course in mathematics, it might be considered case one; whereas in a first year course, discussing the distributive law, it is a very, very bad mistake.
edit: Regarding the comment(s), I'll try to make the argumentation for how many points to give short(er than it was before). When deciding on points for an exercise, you have to see both the whole exam and make sure that points are fair, and you have to look at each single exercise and make sure that it can be properly graded. If you give not enough points, then you might be forced to go down to half points or even quarters when grading, because "There is an error, but still, most is correct...". If you give too many points, you might end up giving points to a completely wrong exercise, because you can only take off so much.
Looking at the exercise you posted, I'd give 4 points.
There are two major tasks here, as explained above: Removing the parentheses and collecting the $x$. For each step, two points is a good number. This will allow to take one point off for minor mistakes (e.g. $5+4 = 8)$ and two points off for major mistakes (e.g. $2x^2 + x^3 = 2x^5$).