Let $a, b, c \in \mathbb{R}$ and $a \neq 0$. To factor the polynomial $ax^2 + bx + c$, we search for $m, n, r, s \in \mathbb{R}$ such that $a = m \cdot r$, $c = n \cdot s$, and $b = m \cdot s + n \cdot r$. If numbers satisfying these conditions can be found, the polynomial is factored as $ax^2 + bx + c = (mx + n)(rx + s)$. Using the method described above, Sude wants to factor the polynomial $2x^2 + bx - 21$ where $b \in \mathbb{R}$. After finding the real numbers $m, n, r$, and $s$ that satisfy the given conditions, she notices that these numbers are each integers. Later, she confuses the places where she should write the numbers $n$ and $s$, and mistakenly factors the polynomial as $(mx + s)(rx + n)$ instead of $(mx + n)(rx + s)$, and finds the factors of the polynomial $2x^2 + x - 21$. Accordingly, what is b? A) 11 B) 12 C) 13 D) 14 E) 15
Let $a, b, c \in \mathbb{R}$ and $a \neq 0$. To factor the polynomial $ax^2 + bx + c$, we search for $m, n, r, s \in \mathbb{R}$ such that $a = m \cdot r$, $c = n \cdot s$, and $b = m \cdot s + n \cdot r$. If numbers satisfying these conditions can be found, the polynomial is factored as $ax^2 + bx + c = (mx + n)(rx + s)$.
Using the method described above, Sude wants to factor the polynomial $2x^2 + bx - 21$ where $b \in \mathbb{R}$. After finding the real numbers $m, n, r$, and $s$ that satisfy the given conditions, she notices that these numbers are each integers. Later, she confuses the places where she should write the numbers $n$ and $s$, and mistakenly factors the polynomial as $(mx + s)(rx + n)$ instead of $(mx + n)(rx + s)$, and finds the factors of the polynomial $2x^2 + x - 21$.
Accordingly, what is b?
A) 11
B) 12
C) 13
D) 14
E) 15