To solve the given equation involving determinants, we will follow these steps:
### Step 1: Write down the determinants
The equation we need to solve is:
\[
|((1+x)^2, (1-x)^2, -(2+x^2)), (2x+1, 3x, 1-5x), (x+1, 2x, 2-3x)| + |((1+x)^2, 2x+1, x+1), ((1-x)^2, 3x, 2x), (1-2x, 3x-2, 2x-3)| = 0
\]
### Step 2: Analyze the first determinant
Let's denote the first determinant as \( D_1 \):
\[
D_1 = |((1+x)^2, (1-x)^2, -(2+x^2)), (2x+1, 3x, 1-5x), (x+1, 2x, 2-3x)|
\]
### Step 3: Analyze the second determinant
Now, denote the second determinant as \( D_2 \):
\[
D_2 = |((1+x)^2, 2x+1, x+1), ((1-x)^2, 3x, 2x), (1-2x, 3x-2, 2x-3)|
\]
### Step 4: Notice the similarity in rows
Observe that the first row of \( D_1 \) and \( D_2 \) are identical:
- First row of \( D_1 \): \((1+x)^2, (1-x)^2, -(2+x^2)\)
- First row of \( D_2 \): \((1+x)^2, 2x+1, x+1\)
### Step 5: Transpose the first determinant
To simplify the calculations, we can take the transpose of the first determinant \( D_1 \):
\[
D_1^T = |(2x + 1, (1+x)^2, (1-x)^2), (3x, 2x+1, 3x), (1-5x, x+1, 2-3x)|
\]
### Step 6: Add the determinants
Now, we can add \( D_1^T \) and \( D_2 \):
\[
D_1^T + D_2 = |(2x + 1 + 2x + 1, (1+x)^2 + (1+x)^2, (1-x)^2 + x+1), (3x + 3x, 2x+1 + 2x, 3x), (1-5x + 1-2x, x+1 + 3x-2, 2-3x)|
\]
### Step 7: Simplify the determinants
After simplification:
- The first two rows of both determinants become identical.
- This indicates that the determinant will equal zero.
### Step 8: Conclusion
Since the first row and the third row of the combined determinant are identical, it implies that the determinant has infinite solutions.
### Final Answer
Thus, the equation has infinite number of solutions, real or non-real. The correct option is:
**(d) infinite number of solutions, real or non-real.**
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