What is the real number m if 49^x+m*7^x-m-1=0 ?

We'll re-write the equation, replacing 7^x by
t;

t^2 + m*t - (m+1) = 0

We'll
consider the cases:

1) delta > 0 and the product of
roots P<0.

2) delta = 0 and the sum of roots
S>0.

delta = b^2 -
4ac

a = 1, b = m and c =
-(m+1)

delta = m^2  +
4(m+1)

delta >=0 => m^2 + 4m + 4 = (m+2)^2
> 0

We'll impose the constraint
P<0

P = x1*x2 = -(m +
1)

-(m + 1) < 0

m +
1> 0

m>-1

We'll
impose the constraint
S>0.

m>0

We'll
impose the constraint delta = 0 => (m+2)^2 => m =
-2.

The intervals of real values of m, such
as 49^x+m*7^x-m-1=0 are: {-2}U(0 , +infinite).