I beg your pardon! I did not notice the condition "has four distinct solutions "

eq:= x^4 -(3*m+2)*x^2 + 3*m+1:

sol:=[solve(eq=0,x)]:

L:=[seq(1..nops(sol))]:

op(map(allvalues,[solve([seq(sol[i]<2, i=1..nops(sol)),seq(seq(`if`(i<>j, sol[i]<>sol[j], NULL),j in L), i in L)],m)]));

                                                {-1/3 < m, m < 0}, {0 < m, m < 1}

@Carl Love

If you want to see the gridlines on the filled region then  transparency  option can be used. 

@Carl Love

If you want to see the gridlines on the filled region then  transparency  option can be used. 

According to Markiyan's idea:

l1:=[a1*t+x1, b1*t+y1, c1*t+z1]:

l2:=[a2*s+x2, b2*s+y2, c2*s+z2]:

minimize(add((l1[i]-l2[i])^2, i=1..3), s=-infinity..infinity, t=-infinity..infinity) assuming a1^2+b1^2+c1^2>0 and a2^2+b2^2+c2^2>0:

RealDomain[simplify](sqrt(%)); 

@Markiyan Hirnyk 

Let the questioner will decide himself which option is more suitable for him. Purely terminological distinction.

@Markiyan Hirnyk 

Let the questioner will decide himself which option is more suitable for him. Purely terminological distinction.

@Markiyan Hirnyk 
The questioner wrote "two positive solutions and two negative solutions". If  m=5/2  then we have one negative and two positive roots.

@Markiyan Hirnyk 
The questioner wrote "two positive solutions and two negative solutions". If  m=5/2  then we have one negative and two positive roots.

@Markiyan Hirnyk 

Of course, I understand that the field is not potential, so the path of integration should be specified.

@Markiyan Hirnyk 

Of course, I understand that the field is not potential, so the path of integration should be specified.

This is a simple task, it is easy to solve by a procedure. But I did not understand the law of getting your matrix for any m.  What is  i ? Can you give an example of such a matrix for a given m, for example, for m = 8 ?

I noticed an error in the text of the procedure. Instead of  if alpha = 0 then {k> 0}  should be  if alpha = 0 then print(`No solutions`) .

@Markiyan Hirnyk

Of course, the advantage of substantially purely aesthetic.
Also, I think this form of writing can be interesting for educational purposes in the study of the concept of absolute value.

PS. The advantage is the same as   |x|  over 

I did not notice the condition that  x>=0, y>=0, z>=0. Therefore, the solution exists.

I did not notice the condition that  x>=0, y>=0, z>=0. Therefore, the solution exists.