The problem asks to find a quadratic equation in the standard form $ax^2 + bx + c = 0$ given the solutions $x = -15$ and $x = -14$, where $a$, $b$, and $c$ are integers with no common factor, and $a$ is positive.

1. Problem Description

The problem asks to find a quadratic equation in the standard form $ax^2 + bx + c = 0$ given the solutions $x = -15$ and $x = -14$, where $a$, $b$, and $c$ are integers with no common factor, and $a$ is positive.

2. Solution Steps

If the solutions to the quadratic equation are $x_1$ and $x_2$, then the equation can be written as

$(x - x_1)(x - x_2) = 0$.

In our case, $x_1 = -15$ and $x_2 = -14$. Substituting these values, we get:

$(x - (-15))(x - (-14)) = 0$

$(x + 15)(x + 14) = 0$

Expanding this expression, we have:

$x^2 + 14x + 15x + (15)(14) = 0$

$x^2 + 29x + 210 = 0$

In this equation, $a = 1$, $b = 29$, and $c = 210$. Since $a$, $b$, and $c$ have no common factors (other than 1) and $a$ is positive, the quadratic equation is $x^2 + 29x + 210 = 0$.

3. Final Answer

$x^2 + 29x + 210 = 0$