To solve the equations for \( x \), let's go through them one by one:

1) \( 4x(3x - 1)^3 - 5^2 = 475 \)
\[
4x(3x - 1)^3 - 25 = 475 \\
4x(3x - 1)^3 = 500
\]
Simplifying might give you potential candidates for \( x \).

2) \( (4x - 3)^3 = 27 \times 125 \)
\[
(4x - 3)^3 = 3375
\]
Take the cube root on both sides:
\[
4x - 3 = 15 \\
4x = 18 \\
x = 4.5
\]

3) \( 2x(23x - 17)^3 = 144 \)
Simplification or substitution might be needed to solve this for \( x \).

4) \( 3^x + 3^{x+1} + 3^{x+2} = -1003 \)
Rewrite using common powers of 3:
\[
3^x + 3 \cdot 3^x + 9 \cdot 3^x = -1003 \\
13 \cdot 3^x = -1003
\]
Divide by 13:
\[
3^x = -77
\]
Since there is no real solution where 3 raised to any power is negative, recheck if there’s an error in the setup.

5) \( 10^{x+1} + 10^x = -11 \)
Factor powers of 10:
\[
10 \cdot 10^x + 10^x = -11 \\
11 \cdot 10^x = -11
\]
\[
10^x = -1
\]
Since 10 raised to any power is positive, real solutions might not be possible.

6) \( 2^{x+3} + 2^x = 36 \)
\[
2^x \cdot 2^3 + 2^x = 36 \\
8 \cdot 2^x + 2^x = 36 \\
9 \cdot 2^x = 36 \\
2^x = 4 \\
x = 2
\]

7) \( 7^{x+2} + 7^{x-2} = 51 \)
Simplify using powers of 7:
\[
49 \cdot 7^x + \frac{1}{2401} \cdot 7^x = 51
\]
Simplify and solve for \( x \).

8) \( 8^{x+1} - 3 \cdot 2^x = -648 \)
\[
8 \cdot 8^x - 3 \cdot 2^x = -648
\]
Represent 8 in terms of powers of 2:
\[
(2^3) \cdot (2^{3x}) - 3(2^x) = -648
\]
Simplify and solve for \( x \).

For each, check the algebra and adjustments during simplification to ensure correct solutions.