After you multiply both sides by $x+3$, you should get an equation where the numerator $4x^3 + kx^2 - 9x - 27$ equals $(x+3)$ times $4x^2 - 9$. Focus on expanding this product correctly.

Once you expand $(x+3)(4x^2 - 9)$, compare it to $4x^3 + kx^2 - 9x - 27$. Look specifically at the coefficient (number in front) of $x^2$ to determine $k$.

In Desmos, type k = 0 on a new line to create a slider for $k$. On the next line, enter (4x^3 + kx^2 - 9x - 27)/(x + 3) to graph the rational expression.

On another line, type 4x^2 - 9 to graph the quadratic expression.

Move the $k$ slider to each of the answer choice values one at a time. For the correct value of $k$, the graph of (4x^3 + kx^2 - 9x - 27)/(x + 3) will lie exactly on top of the graph of 4x^2 - 9 for all $x$ except at $x = -3$, where the rational expression is undefined.

We are told

for all $x \ne -3$. Since $x+3 \ne 0$ for these $x$, we can multiply both sides by $x+3$:

Now the problem becomes: find $k$ so that this polynomial equality is true for all $x$.

Expand $(x+3)(4x^2 - 9)$ by distributing:

First distribute $x$:

Then distribute $3$:

Add these results:

From Step 1, we need

For two polynomials to be equal for all $x$, the coefficients (the numbers in front of each power of $x$) must match term by term. The $x^2$ term on the left is $kx^2$, and on the right it is $12x^2$, so we must have

Therefore, the correct answer is 12 (choice C).