To evaluate the expressions \( E \) and \( G \), we'll simplify each step-by-step.

### Expression \( E \):
\[ E = \sqrt{9 - 4\sqrt{5}} + \sqrt{6 + 2\sqrt{5}} \]

For terms like \(\sqrt{a \pm b\sqrt{c}}\), try assuming they can be written in the form \(\sqrt{x} \pm \sqrt{y}\):

#### \(\sqrt{9 - 4\sqrt{5}}\):
Assume \(\sqrt{a} - \sqrt{b} = \sqrt{9 - 4\sqrt{5}}\).

From \((\sqrt{x} - \sqrt{y})^2 = x + y - 2\sqrt{xy}\), we compare:
\[ x + y = 9 \]
\[ -2\sqrt{xy} = -4\sqrt{5} \]
\[ \Rightarrow xy = 5 \]

Solving the equations:
- \( x + y = 9 \)
- \( xy = 5 \)

Solve the system:
\[ t^2 - 9t + 5 = 0 \]

Using the quadratic formula:
\[ t = \frac{9 \pm \sqrt{81 - 20}}{2} = \frac{9 \pm \sqrt{61}}{2} \]

Values are \( x = \frac{9+\sqrt{61}}{2}, y = \frac{9-\sqrt{61}}{2} \).

#### \(\sqrt{6 + 2\sqrt{5}}\):
Assume \(\sqrt{a} + \sqrt{b} = \sqrt{6 + 2\sqrt{5}}\).

Comparing:
\[ x + y = 6 \]
\[ 2\sqrt{xy} = 2\sqrt{5} \]
\[ \Rightarrow xy = 5 \]

Solving similarly:
\[ t^2 - 6t + 5 = 0 \]

Solving gives:
\[ t = \frac{6 \pm \sqrt{36 - 20}}{2} = \frac{6 \pm \sqrt{16}}{2} = \frac{6 \pm 4}{2} \]

Values are \( x = 5, y = 1 \).

### Expression \( G \):
\[ G = \sqrt{12 + 8\sqrt{2}} + \sqrt{6 - 4\sqrt{2}} \]

#### \(\sqrt{12 + 8\sqrt{2}}\):
Assume \(\sqrt{x} + \sqrt{y} = \sqrt{12 + 8\sqrt{2}}\).

Comparing:
\[ x + y = 12 \]
\[ 2\sqrt{xy} = 8\sqrt{2} \]
\[ \Rightarrow xy = 32 \]

Solve:
\[ t^2 - 12t + 32 = 0 \]

Using the quadratic formula:
\[ t = \frac{12 \pm \sqrt{144 - 128}}{2} = \frac{12 \pm \sqrt{16}}{2} = \frac{12 \pm 4}{2} \]

Values are \( x = 8, y = 4 \).

#### \(\sqrt{6 - 4\sqrt{2}}\):
Assume \(\sqrt{x} - \sqrt{y} = \sqrt{6 - 4\sqrt{2}}\).

Comparing:
\[ x + y = 6 \]
\[ -2\sqrt{xy} = -4\sqrt{2} \]
\[ \Rightarrow xy = 8 \]

Solve:
\[ t^2 - 6t + 8 = 0 \]

Using the quadratic formula:
\[ t = \frac{6 \pm \sqrt{36 - 32}}{2} = \frac{6 \pm \sqrt{4}}{2} = \frac{6 \pm 2}{2} \]

Values are \( x = 4, y = 2 \).

Putting it all together for the simplified solutions, combine the evaluations for \( E \) and \( G \).