Given that [tex]tga = 2[/tex], we can substitute [tex]tga[/tex] with its equivalent value:

[tex]tga = \frac{sina}{cosa} = 2[/tex]

From this, we can rearrange to solve for [tex]sina[/tex]:

[tex]sina = 2cosa[/tex]

Now, we can substitute [tex]sina = 2cosa[/tex] back into the expression:

[tex]\frac{3cosa - 4(2cosa)}{4(2cosa) + 2cosa}[/tex]

Simplify further:

[tex]\frac{3cosa - 8cosa}{8cosa + 2cosa}[/tex]

[tex]\frac{-5cosa}{10cosa}[/tex]

[tex]\frac{-5}{10}[/tex]

[tex]-\frac{1}{2}[/tex]

Therefore, the simplified value of the expression is [tex]-\frac{1}{2}[/tex].