1. **Stating the problem:** We want to find the term independent of $x$ (the constant term) in the expansion of $$\left(2 + \frac{3}{x^2}\right)^{10} \left(1 - 4x^2\right)^2.$$\n\n2. **Expand each part separately:**\n(a) For $$\left(2 + \frac{3}{x^2}\right)^{10},$$ use the binomial theorem: $$\sum_{k=0}^{10} \binom{10}{k} 2^{10-k} \left(\frac{3}{x^2}\right)^k = \sum_{k=0}^{10} \binom{10}{k} 2^{10-k} 3^k x^{-2k}.$$\n(b) For $$\left(1 - 4x^2\right)^2,$$ expand directly: $$1 - 8x^2 + 16x^4.$$\n\n3. **Multiply the expansions and look for the constant term:**\nThe general product term combining \(k\) from the first sum and the power terms from the second expansion is:\n$$ \binom{10}{k} 2^{10-k} 3^k x^{-2k} \times \{1, -8x^2, 16x^4\}.$$\n\n4. **Find combinations where powers of $x$ cancel out to zero:**\nThe power of $x$ in a product term is $$-2k + m,$$ where $m$ is $0$, $2$, or $4$ corresponding to the terms $1$, $-8x^2$, and $16x^4$ respectively. We want\n$$-2k + m = 0.$$\n\nCheck each $m$:\n- For $m=0$: $$-2k = 0 \implies k=0.$$\n- For $m=2$: $$-2k + 2 = 0 \implies k=1.$$\n- For $m=4$: $$-2k + 4 = 0 \implies k=2.$$\n\n5. **Calculate the constant term contributions for each valid $k$: **\n- $k=0$, $m=0$ (term from first: $\binom{10}{0}2^{10}3^0=1 \times 1024 \times 1=1024$, from second: $1$) \n Contribution: $1024 \times 1 = 1024$\n- $k=1$, $m=2$ (term from first: $\binom{10}{1}2^{9}3^{1}=10 \times 512 \times 3=15360$, from second: $-8x^2$) \n Contribution: $15360 \times (-8) = -122880$\n- $k=2$, $m=4$ (term from first: $\binom{10}{2}2^{8}3^{2} = 45 \times 256 \times 9=103680$, from second: $16x^4$) \n Contribution: $103680 \times 16 = 1658880$\n\n6. **Sum all contributions:**\n$$1024 - 122880 + 1658880 = 1539024.$$\n\n**Final answer:** The exact value of the term independent of $x$ is **$1539024$.**