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i am lost. where does this line come from
\[6x + 22 = Ax + 2A + Bx - 4B\]

oh nvm i see the mistake

6x + 22/(x+2)(x-4) dx

What is the mistake?

Let x = -2. Then, 6(2) + 22 = A(-2 + 2) + B(-2 - 4)

should be Let x = -2.
Then, 6(-2) + 22 = A(-2 + 2) + B(-2 - 4)

\[a+b=6\]
\[a+2b=11\]
\[b=5\]

What am I doing wrong wrt the trick? Why am I not getting the same answer as the second method?

i see mistake when you put 6(2) in place of 6(-2) ....in first trick

did you see that?

ok lets start at the top
is it
\[\frac{6x + 22 }{(x+2)(x-4)}=\frac{a}{x+2}+\frac{b}{x-4}\]

that's right

Then why is it NOT the same as "No trick"?

a+b=6 is same as 2(a+b)=2(6) is same as 2a+2b=12 is same as -2a -2b =-12
add -2a-2b=-12 and 2a-4b=22

Alright, then how about the "No trick"?

typo on last line
should be
\[\frac{6x+22}{(x+2)(x-4)}=\frac{23}{3(x+2)}-\frac{5}{3(x-4)}\]

add left left and right right
-2a-2b=-12 and 2a-4b=22 gives -6b=10 so b is -10/6 same as first one

i get the same answer solving this one

yes.....