Factorize the Expression: a²b²c² - a²b⁵c⁴ + a⁻⁴b³c⁷

Decompose the following expression into factors:

$a^2b^2c^2-a^2b^5c^4+a^{-4}b^3c^7$

To solve this problem, let's perform the factorization step by step:

We start with the expression:

Step 1: Identify the common factors among terms:

• The smallest exponent of $a$ among terms with positive exponents is $2$, excluding any negative exponent (as considering $a^{-4}$ would introduce complexity).
• The smallest exponent of $b$ is $2$.
• The smallest exponent of $c$ is $2$.

Therefore, a common factor considering positive indices is:

Step 2: Factor out $a^2b^2c^2$ from the entire expression:

Simplifying the terms inside the parenthesis, we have:

• $\frac{a^2b^2c^2}{a^2b^2c^2} = 1$ (since dividing any expression by itself gives 1).
• $\frac{a^2b^5c^4}{a^2b^2c^2} = b^3c^2$.
• $\frac{a^{-4}b^3c^7}{a^2b^2c^2} = a^{-6}bc^5$.

Putting it all together inside the parenthesis, we get:

Thus, the completely factored expression is:

Verifying against given choices, this matches exactly with the correct choice:

$a^2b^2c^2\left(a-b^3c^3+a^{-6}bc^5\right)$

Therefore, the solution to the problem is $\boxed{a^2b^2c^2(a-b^3c^3+a^{-6}bc^5)}$.