The problem asks to factor the quadratic expression $15x^2 - 2x + 3$, if possible. If the polynomial cannot be factored, we say it is prime.

1. Problem Description

The problem asks to factor the quadratic expression $15x^2 - 2x + 3$, if possible. If the polynomial cannot be factored, we say it is prime.

2. Solution Steps

We need to determine if the quadratic expression $15x^2 - 2x + 3$ can be factored.

We look for two binomials of the form $(ax + b)(cx + d)$ such that their product is equal to $15x^2 - 2x + 3$.

The product of the first terms, $ax * cx$, must be $15x^2$, so $ac = 15$.

The product of the last terms, $b * d$, must be $3$.

The sum of the outer and inner terms, $ax * d + bx * c$, must be $-2x$, so $ad + bc = -2$.

Possible pairs for $a$ and $c$ are (1, 15), (3, 5), (5, 3), (15, 1).

Possible pairs for $b$ and $d$ are (1, 3), (3, 1), (-1, -3), (-3, -1).

Let's try $a = 3$ and $c = 5$.

If $b = 1$ and $d = 3$, then $ad + bc = 3(3) + 1(5) = 9 + 5 = 14$, which is not $-2$.

If $b = 3$ and $d = 1$, then $ad + bc = 3(1) + 3(5) = 3 + 15 = 18$, which is not $-2$.

If $b = -1$ and $d = -3$, then $ad + bc = 3(-3) + (-1)(5) = -9 - 5 = -14$, which is not $-2$.

If $b = -3$ and $d = -1$, then $ad + bc = 3(-1) + (-3)(5) = -3 - 15 = -18$, which is not $-2$.

Let's consider the discriminant, which can determine if the quadratic equation has real roots.

The discriminant is given by the formula:

$D = b^2 - 4ac$

In our case, $a = 15$, $b = -2$, and $c = 3$.

$D = (-2)^2 - 4(15)(3) = 4 - 180 = -176$.

Since the discriminant is negative, the quadratic equation has no real roots. This implies that the quadratic expression cannot be factored using real numbers.

3. Final Answer

B. The polynomial is prime