To solve the equations \( x^2 - 8 = 43 \) and \( x^2 - 8 = 4 \), we'll handle them one at a time.

### 1st Equation: \( x^2 - 8 = 43 \)

1. First, add 8 to both sides:
\[
x^2 = 43 + 8
\]
\[
x^2 = 51
\]

2. Now take the square root of both sides:
\[
x = \pm \sqrt{51}
\]

### 2nd Equation: \( x^2 - 8 = 4 \)

1. Again, add 8 to both sides:
\[
x^2 = 4 + 8
\]
\[
x^2 = 12
\]

2. Take the square root of both sides:
\[
x = \pm \sqrt{12}
\]
\[
x = \pm 2\sqrt{3}
\]

### Summary of Solutions

1. For the equation \( x^2 - 8 = 43 \), the solutions are:
\[
x = \pm \sqrt{51}
\]

2. For the equation \( x^2 - 8 = 4 \), the solutions are:
\[
x = \pm 2\sqrt{3}
\]