should I use double negative in a compound sentence?

Question

The following two sentences are compound having negative expression-

They don't learn English at all and reach the target stage.

They can not write English sentence correctly and speak out their feelings in that language.

I found them from one of my text books,but it's seeming really horrible to me as there is no use of negative word in the second clause of both of the sentences but still they want to express a pure negative state.
Now say me,will I adopt this structure or use negative word in the second clause?I think,I should opt for the second one because at times I may need to express positive state in the second clause,then it will be difficult to distinguish the meaning from each other.

Answer 1

You're quite right: as it stands, the second clause implies that as a result of failing to learn English they do reach the 'target stage'!

This is a very tricky use of negatives with ellipsis. What the writer intends is something like this:

  They don't     learn English at all
 and
 (they don't)    reach the target stage. 

But this doesn't work. The expression at all is what linguists call a 'Negative Polarity Item': it is only used in negative contexts. In this sentence it combines with the not in don't and so to speak "absorbs" its negative sense before you get to the second clause. Consequently, that not cannot take the second clause into its scope. The parse looks something like this

   They    [do not at all] learn English   
  and 
  (they)                   reach the second stage 

You have a number of options for correcting this. One is what you suggest: adding a negative expression to the second clause. This Since the negative in the first clause is comprehensive, not at all, a good choice for the second clause would be the equally comprehensive never:

They don't learn English at all and never reach the target stage.

That would be parsed something like this:

   They  [do not at all] learn English   
  and 
  (they) [    never    ] reach the target stage 

Another option is that suggested by snailboat: employ the 'exclusive' coordinator or instead of and, and pair the NPI at all with a corresponding NPI in the second clause

They don't learn English at all or ever reach the target stage.

   They don't [at all] learn English   
           or 
              [ ever ] reach the target stage 

Finally, you could coordinate learn English with reach the target stage using or, and move the NPI at all to a point where it modifies the coordination:

They don't learn English or reach the target stage at all.

            (learn English         )
 They don't ( or                   )   at all.
            (reach the target stage)

ADDDED:
Your second sentence, "They can not write English sentence correctly and speak out their feelings in that language", is marginally acceptable; there's no ambiguity about the scope of the negator. But it would be better expressed with or, which alerts the reader to the fact that what follows falls under the scope of they cannot:

They cannot write an English sentence correctly or speak out their feelings in that language.