To determine which polynomial is prime, let's analyze each polynomial given:
1. [tex]\( 8x^2 - 10x - 3 \)[/tex]
2. [tex]\( 8x^2 + 2x - 3 \)[/tex]
3. [tex]\( 8x^2 - 6x - 3 \)[/tex]
4. [tex]\( 8x^2 + 23x - 3 \)[/tex]
A prime polynomial cannot be factored into the product of two or more non-constant polynomials with coefficients in the same set (usually integers).
Let's check each polynomial one by one:
1. Polynomial: [tex]\( 8x^2 - 10x - 3 \)[/tex]
After factoring, this polynomial can be expressed as:
[tex]\[
8x^2 - 10x - 3 = (2x - 3)(4x + 1)
\][/tex]
Since it can be written as a product of two polynomials of lower degrees, it is not a prime polynomial.
2. Polynomial: [tex]\( 8x^2 + 2x - 3 \)[/tex]
After factoring, this polynomial can be expressed as:
[tex]\[
8x^2 + 2x - 3 = (2x - 1)(4x + 3)
\][/tex]
Since it can be written as a product of two polynomials of lower degrees, it is not a prime polynomial.
3. Polynomial: [tex]\( 8x^2 - 6x - 3 \)[/tex]
This polynomial cannot be factored further into polynomials of lower degrees with integer coefficients. Hence:
[tex]\[
8x^2 - 6x - 3 \quad \text{(cannot be expressed as a product of lower-degree polynomials)}
\][/tex]
Therefore, this is a prime polynomial.
4. Polynomial: [tex]\( 8x^2 + 23x - 3 \)[/tex]
After factoring, this polynomial can be expressed as:
[tex]\[
8x^2 + 23x - 3 = (x + 3)(8x - 1)
\][/tex]
Since it can be written as a product of two polynomials of lower degrees, it is not a prime polynomial.
Based on the above analysis, the polynomial that is prime is:
[tex]\[
\boxed{8x^2 - 6x - 3}
\][/tex]
