We have -3 + sqrt(m+59) = m to solve for m

-3 + sqrt(m+59) = m

=> sqrt (m + 59) = 3 + m

square both the sides

=> m + 59 = 9 + m^2 + 6m

=> m^2 + 5m - 50 = 0

=> m^2 + 10m - 5m - 50 = 0

=> m(m + 10) - 5(m + 10) = 0

=> (m - 5)(m + 10) = 0

=> m = 5 and m = -10

For m = -10 we get 4 = -10, so we eliminate this root.

**The value of m is 5.**

We'll start by imposing the condition of existence of the square root.

m +59 >=0

m >= -59

The range of admissible values of m are [-59 ; +infinite)

Now, we'll solve the equation:

-3 + sqrt(m+59) = m

sqrt(m+59) = m + 3

We'll raise to square both sides:

m + 59 = (m+3)^2

m + 59 = m^2 + 6m + 9

We'll move all terms to the right side and we'll use the symmetric property:

m^2 + 6m + 9 - m - 59 = 0

We'll combine like terms:

m^2 + 5m - 50 = 0

We'll apply the quadratic formula:

m1 = [-5+sqrt(25 + 200)]/2

m1 = (-5+15)/2

m1 = 5

m2 = (-5-15)/2

m2 = -10

Since both values belong to interval of admissible values, they are accepted.