Simplify:
`(2a-3b+1)^(2)+(2a-3b-1)^(2)`

Answer

To simplify the expression \((2a-3b+1)^{2}+(2a-3b-1)^{2}\), we will use the algebraic identity for the square of a binomial.

### Step-by-Step Solution:

1. **Identify the terms**:
Let \(x = 2a - 3b\). Then we can rewrite the expression as:
\[ (x + 1)^{2} + (x - 1)^{2} \]

2. **Apply the binomial square identity**:
The identity states that \((a + b)^{2} = a^{2} + 2ab + b^{2}\). Thus, we can expand both squares:
\[ (x + 1)^{2} = x^{2} + 2x(1) + 1^{2} = x^{2} + 2x + 1 \]
\[ (x - 1)^{2} = x^{2} - 2x(1) + 1^{2} = x^{2} - 2x + 1 \]

3. **Combine the expanded forms**:
Now, add the two expanded forms together:
\[ (x + 1)^{2} + (x - 1)^{2} = (x^{2} + 2x + 1) + (x^{2} - 2x + 1) \]

4. **Simplify the expression**:
Combine like terms:
\[ = x^{2} + x^{2} + 2x - 2x + 1 + 1 \]
\[ = 2x^{2} + 2 \]

5. **Substitute back for \(x\)**:
Recall that \(x = 2a - 3b\):
\[ = 2(2a - 3b)^{2} + 2 \]

6. **Expand \( (2a - 3b)^{2} \)**:
\[ (2a - 3b)^{2} = (2a)^{2} - 2(2a)(3b) + (3b)^{2} = 4a^{2} - 12ab + 9b^{2} \]

7. **Combine everything**:
Substitute back into the expression:
\[ = 2(4a^{2} - 12ab + 9b^{2}) + 2 \]
\[ = 8a^{2} - 24ab + 18b^{2} + 2 \]

### Final Answer:
Thus, the simplified expression is:
\[ \boxed{8a^{2} - 24ab + 18b^{2} + 2} \]