Use the binomial theorem to find the term containing $x^{12}$ in the expansion $(-4 x-4)^{15}$

Step 1 ：Given the binomial expansion \((a+b)^n = \sum_{k=0}^{n} {n \choose k} a^{n-k} b^k\), we are asked to find the term containing $x^{12}$ in the expansion \((-4 x-4)^{15}\). Here, $a = -4x$, $b = -4$, and $n = 15$.

Step 2 ：We want to find the term where the power of $x$ is 12, which means $n-k = 12$. Therefore, $k = n - 12 = 15 - 12 = 3$. So, we need to find the coefficient of the term where $k = 3$.

Step 3 ：Substituting $n = 15$, $k = 3$, $a = -4$, and $b = -4$ into the binomial theorem, we find that the coefficient is -488552529920.

Step 4 ：Thus, the term containing $x^{12}$ in the expansion \((-4 x-4)^{15}\) is \(\boxed{-488552529920x^{12}}\).