Factor. \[ -27 a^{4}+144 a^{5}-192 a^{6} \] \( -27 a^{4}+144 a^{5}-192 a^{6}=\square \) (Factor completely.

First, let's factor out the greatest common factor (GCF) from the expression. The GCF is \( -27a^4 \), so we have: \[ -27a^4(1 - \frac{144}{-27}a + \frac{192}{-27}a^2) \] This simplifies to: \[ -27a^4(1 - \frac{144}{27}a + \frac{192}{27}a^2) = -27a^4(1 - \frac{16}{3}a + \frac{64}{9}a^2) \] Next, we can factor the quadratic expression inside the parentheses. The expression \( 1 - \frac{16}{3}a + \frac{64}{9}a^2 \) can be factored using the quadratic formula or by recognizing it as a perfect square: \[ 1 - \frac{16}{3}a + \left(\frac{8}{3}a\right)^2 = \left(1 - \frac{8}{3}a\right)^2 \] Thus, we get: \[ -27a^4 \left(1 - \frac{8}{3}a\right)^2 \] Putting it all together, the completely factored form is: \[ -27a^4\left(1 - \frac{8}{3}a\right)^2 \]