The relationship between the rhs and the lhs is that of an identity.
The rhs has to be identical to the lhs for all values of x.
Take then your equation Ax + Bx + A - 2B = x + 7,
strictly it should be written
\(\displaystyle Ax + Bx + A - 2B \equiv x +7.\)
For the two to be identical, (producing identical results for all values of x), it's necessary that the number of x's on the lhs should be the same as the number of x's on the rhs and also that the constants should be the same.
That leads to two equations,
\(\displaystyle A+B=1 \text{ and } A-2B=7, \text{ from which } A=3 \text{ and } B=-2.\)
Check then to see that
\(\displaystyle \frac{3}{x-2}-\frac{2}{x+1}\equiv \frac{x+7}{x^{2}-x-2}.\)
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