To solve the equation \((x-3)^2 - 5 = 4^2 + 2^2\), let's follow these steps:

1. Simplify the right-hand side of the equation.
2. Solve for \(x\).

First, simplify the right-hand side:

\[ 4^2 + 2^2 = 16 + 4 = 20 \]

So the equation becomes:

\[ (x-3)^2 - 5 = 20 \]

Next, add 5 to both sides to isolate the squared term:

\[ (x-3)^2 - 5 + 5 = 20 + 5 \]

\[ (x-3)^2 = 25 \]

Now, take the square root of both sides:

\[ \sqrt{(x-3)^2} = \pm \sqrt{25} \]

\[ x-3 = \pm 5 \]

This gives us two possible solutions:

1. \( x-3 = 5 \)
2. \( x-3 = -5 \)

Solve each equation for \(x\):

1. \( x-3 = 5 \)

\[ x = 5 + 3 \]

\[ x = 8 \]

2. \( x-3 = -5 \)

\[ x = -5 + 3 \]

\[ x = -2 \]

So, the solutions to the equation \((x-3)^2 - 5 = 4^2 + 2^2\) are:

\[ x = 8 \quad \text{and} \quad x = -2 \]