find real numbers a and b if
a(3x+1)+b(5x-1)=11x+9 for all x€r
you have
3ax + a + 5bx - b = 11x + 9
If these expressions are identical for all x, then like powers of x must be equal:
3a + 5b = 11
a - b = 9
so,
3(b+9) + 5b = 11
b = -2
a = 7