To identify the equivalent four-term polynomial using the X method, we will follow these steps for each polynomial:
1. Simplify each polynomial by combining like terms.
2. Identify the unique simplified polynomial.
Let's examine each provided polynomial one by one:
1. For the polynomial [tex]\(x^2 + 8x + 8x + 48\)[/tex]:
- Combine the like terms: [tex]\(8x + 8x\)[/tex] becomes [tex]\(16x\)[/tex].
- Result: [tex]\(x^2 + 16x + 48\)[/tex].
2. For the polynomial [tex]\(x^2 - 12x - 4x + 48\)[/tex]:
- Combine the like terms: [tex]\(-12x - 4x\)[/tex] becomes [tex]\(-16x\)[/tex].
- Result: [tex]\(x^2 - 16x + 48\)[/tex].
3. For the polynomial [tex]\(x^2 + 12x + 4x + 48\)[/tex]:
- Combine the like terms: [tex]\(12x + 4x\)[/tex] becomes [tex]\(16x\)[/tex].
- Result: [tex]\(x^2 + 16x + 48\)[/tex].
4. For the polynomial [tex]\(x^2 - 8x - 8x + 48\)[/tex]:
- Combine the like terms: [tex]\(-8x - 8x\)[/tex] becomes [tex]\(-16x\)[/tex].
- Result: [tex]\(x^2 - 16x + 48\)[/tex].
From simplification:
- The polynomial [tex]\(x^2 + 16x + 48\)[/tex] appears in the first and third options.
- The polynomial [tex]\(x^2 - 16x + 48\)[/tex] appears in the second and fourth options.
Identifying the unique simplified polynomial, we find that [tex]\(x^2 + 16x + 48\)[/tex] is the simplified form that fits with the first given polynomial, [tex]\(x^2 + 8x + 8x + 48\)[/tex].
Hence, the equivalent four-term polynomial that can be created using the X method and is simplified to the unique form [tex]\(x^2 + 16x + 48\)[/tex] corresponds to the first option.
Thus, the answer is:
[tex]\[ \boxed{1} \][/tex]
