Filling the Gap: Determine the Unknown Coefficient to Make -1 a Solution

To solve this problem, we'll follow these steps:

• Step 1: Substitute the given solution $x = -1$ into the equation.
• Step 2: Simplify the equation and solve for the missing coefficient.

Now, let's work through each step:

Step 1: Substitute $x = -1$ into the equation $7x^2 + \square x + 2 = 0$.
This gives us:
$7(-1)^2 + \square(-1) + 2 = 0$.

Step 2: Simplify the equation.
We know that $(-1)^2 = 1$, so:
$7 \times 1 - \square + 2 = 0$.

This simplifies to:
$7 - \square + 2 = 0$,
which simplifies further to:
$9 - \square = 0$.

Solving for the blank, we have:
$\square = 9$.

Therefore, the missing coefficient is $9$.