To solve the given equations:

1) x + 6 = x^2

Rearrange the equation to set it equal to zero:
x^2 - x - 6 = 0

Now, factor the quadratic equation:
$x-3$$x+2$ = 0

So, x = 3 or x = -2.

2) x^2 + 4 = 5x

Rearrange the equation to set it equal to zero:
x^2 - 5x + 4 = 0

Now, factor the quadratic equation:
$x-4$$x-1$ = 0

So, x = 4 or x = 1.

3) $x-3$$x+3$ = -5x - 13

Expand the left side of the equation:
x^2 - 9 = -5x - 13

Rearrange the equation:
x^2 + 5x - 4 = 0

Now, factor the quadratic equation:
$x+4$$x-1$ = 0

So, x = -4 or x = 1.