To solve each equation listed, we can start with them one at a time. Here's a step-by-step approach for each equation:

### 1. \((3x - 4)^2 - (x + 2)^2 = 0\)

This can be factored using the difference of squares:

\[
(3x - 4 - (x + 2))(3x - 4 + (x + 2)) = 0
\]

Simplify each factor:

1. \(3x - 4 - x - 2 = 2x - 6 = 2(x - 3) = 0 \Rightarrow x = 3\)
2. \(3x - 4 + x + 2 = 4x - 2 = 0 \Rightarrow x = \frac{1}{2}\)

**Solutions:** \( x = 3, \frac{1}{2} \)

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### 2. \((2x - 1)^2 - (3 - x)^2 = 0\)

Using the difference of squares again:

\[
(2x - 1 - (3 - x))(2x - 1 + (3 - x)) = 0
\]

Simplify each factor:

1. \(2x - 1 - 3 + x = 3x - 4 = 0 \Rightarrow x = \frac{4}{3}\)
2. \(2x - 1 + 3 - x = x + 2 = 0 \Rightarrow x = -2\)

**Solutions:** \( x = \frac{4}{3}, -2 \)

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### 3. \((3x + 1)^2 - 16 = 0\)

Rearranging:

\[
(3x + 1)^2 = 16
\]
Taking the square root of both sides:

\[
3x + 1 = \pm 4
\]

Solving these:

1. \(3x + 1 = 4 \Rightarrow 3x = 3 \Rightarrow x = 1\)
2. \(3x + 1 = -4 \Rightarrow 3x = -5 \Rightarrow x = -\frac{5}{3}\)

**Solutions:** \( x = 1, -\frac{5}{3} \)

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### 4. \((x + 3)^2 - x^2 = 45\)

Rearranging gives:

\[
(x + 3)^2 = x^2 + 45
\]

Expanding the left side:

\[
x^2 + 6x + 9 = x^2 + 45
\]

Subtract \(x^2\) from both sides:

\[
6x + 9 - 45 = 0 \Rightarrow 6x - 36 = 0 \Rightarrow 6x = 36 \Rightarrow x = 6
\]

**Solution:** \( x = 6 \)

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### Summary of Solutions
1. \( x = 3, \frac{1}{2} \)
2. \( x = \frac{4}{3}, -2 \)
3. \( x = 1, -\frac{5}{3} \)
4. \( x = 6 \)