We are given the following equation:

$x^2 + mx + n \equiv (x + 2)(x - m) - 8$

Our goal is to determine the value of $n$.

Step 1: Expand the right-hand side

Let's first expand the right-hand side of the equation:

$(x + 2)(x - m) - 8$

Use the distributive property (FOIL method) to expand $(x + 2)(x - m)$:

$(x + 2)(x - m) = x(x - m) + 2(x - m) = x^2 - mx + 2x - 2m$

So, the expression becomes:

$x^2 - mx + 2x - 2m - 8$

Step 2: Equate the two sides

We are told that:

$x^2 + mx + n \equiv x^2 - mx + 2x - 2m - 8$

Now, equate the coefficients of the terms on both sides.

• The $x^2$-terms: Both sides have $x^2$, so they match.
• The $x$-terms: On the left side, the coefficient of $x$ is $m$, and on the right side, the coefficient of $x$ is $-m + 2$. Hence:

$m = -m + 2$

Solving for $m$:

$2m = 2 \quad \Rightarrow \quad m = 1$

Step 3: Solve for $n$

Now that we know $m = 1$, substitute this value into the equation for the constant terms.

On the left side, the constant term is $n$. On the right side, after substituting $m = 1$, the constant term becomes:

$-2m - 8 = -2(1) - 8 = -2 - 8 = -10$

So, equating the constants:

$n = -10$

Final Answer:

The value of $n$ is $\boxed{-10}$, which corresponds to option A.

Do you have any questions or need more details? Here are some related questions to consider:

1. How do you apply the distributive property to expressions like $(x + 2)(x - m)$?
2. Why is it necessary to match the coefficients of $x$ when solving equations like this?
3. What other methods can be used to solve quadratic equations in this form?
4. How does substituting values simplify algebraic expressions?
5. Can the result $m = 1$ be confirmed using a different approach?

Tip: Always check that the coefficients and constant terms on both sides of the equation match after expanding and simplifying!