Is there a number a such that the following limit exists? (If an answer does not exist, enter DNE.) lim x→−2 4x2 + ax + a + 12x2 + x − 2 Find the value a.

To determine if there exists a number $a$ such that the following limit exists:

$\lim_{x \to -2} \frac{4x^2 + ax + a + 12x^2 + x - 2}{g(x)}$

First, we need to clarify what $g(x)$ is. Assuming $g(x)$ is some polynomial and is not given, we usually consider if the expression in the limit approaches indeterminate form, such as $\frac{0}{0}$ when substituting $x = -2$.

Let's first set the numerator $N(x) = 4x^2 + ax + a + 12x^2 + x - 2$ and find the conditions under which $N(-2) = 0$ to ensure the limit exists.

Step 1: Substitute $x = -2$ into $N(x)$

Substituting $x = -2$:

$N(-2) = 4(-2)^2 + a(-2) + a + 12(-2)^2 + (-2) - 2$

Calculating this step-by-step:

1. $4(-2)^2 = 4 \cdot 4 = 16$
2. $a(-2) = -2a$
3. $12(-2)^2 = 12 \cdot 4 = 48$
4. Combine all the terms:

$N(-2) = 16 - 2a + a + 48 - 2 = 0$ $N(-2) = 62 - a = 0$

Step 2: Solve for $a$

Setting the equation to zero:

$62 - a = 0$ $a = 62$

Step 3: Verify and Summarize

When $a = 62$, substituting back gives us a polynomial $N(x)$ that is defined at $x = -2$ resulting in an existence of the limit.

Final Answer

The value of $a$ such that the limit exists is:

$\boxed{62}$