Given the equation -3 + sqrt(m+ 59) = m.

We need to find the values of m.

Let us solve.

-3 + sqrt(m+59) = m

We will add 3 to both sides.

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

Now we will square both sides.

==> m+ 59 = (m+3)^2.

Now let us open the brackets.

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

Now we will combine like terms.

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

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

Now we will factor.

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

==>** m1= -10**

**==> m2= 5**

**Then, there are two values for m that satisfies the equation.**

**==> m = { -10, 5}**

First, we'll impose the condition of existence of the square root.

m +59 >=0

m >= -59

The interval 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 are in the interval of admissible values, they are accepted.