Let's evaluate each of the given expressions step by step.

### a) \(6^2 : 4 \times 3 + 2 \times (-5)^2\)

1. Calculate \(6^2\):
\[
6^2 = 36
\]

2. Calculate \(4 \times 3\):
\[
4 \times 3 = 12
\]

3. Now perform the division:
\[
36 : 12 = 3
\]

4. Calculate \((-5)^2\):
\[
(-5)^2 = 25
\]

5. Calculate \(2 \times 25\):
\[
2 \times 25 = 50
\]

6. Now add the results:
\[
3 + 50 = 53
\]

So, the result for expression **a** is:
\[
\boxed{53}
\]

### b) \(15x^2 + 4x^3 - 5x^7\)

This expression is already simplified and is in polynomial form. Therefore, it cannot be simplified further without specific values for \(x\). The result is simply:
\[
\boxed{15x^2 + 4x^3 - 5x^7}
\]

### c) \(150 - [10^2 - (14 - 11)^2 - 117^0]\)

1. Calculate \(10^2\):
\[
10^2 = 100
\]

2. Calculate \(14 - 11\):
\[
14 - 11 = 3
\]

3. Calculate \((3)^2\):
\[
3^2 = 9
\]

4. Calculate \(117^0\):
\[
117^0 = 1 \quad (\text{Any non-zero number to the power of 0 is 1})
\]

5. Substitute back into the brackets:
\[
150 - [100 - 9 - 1]
\]

6. Simplify inside the brackets:
\[
100 - 9 = 91
\]
\[
91 - 1 = 90
\]

7. Now calculate:
\[
150 - 90 = 60
\]

So, the result for expression **c** is:
\[
\boxed{60}
\]

### Summary of Results:
- a) \(53\)
- b) \(15x^2 + 4x^3 - 5x^7\)
- c) \(60\)