The given expression is [(a + 4b)^3 * (a + 4b)^3]^2.

To find the number of terms in the expansion, we need to determine the number of terms in each of the individual expressions (a + 4b)^3 and [(a + 4b)^3 * (a + 4b)^3].

1. (a + 4b)^3:
The binomial expansion of (a + 4b)^3 can be found using the binomial theorem, which states that for any positive integer n:

(a + b)^n = nC0 * a^n * b^0 + nC1 * a^(n-1) * b^1 + nC2 * a^(n-2) * b^2 + ... + nCn * a^0 * b^n,

where nCr represents the number of combinations of n items taken r at a time.

In this case, we have (a + 4b)^3, so n = 3. Plugging the values into the binomial theorem formula, we get:

(a + 4b)^3 = 3C0 * a^3 * (4b)^0 + 3C1 * a^2 * (4b)^1 + 3C2 * a^1 * (4b)^2 + 3C3 * a^0 * (4b)^3.

Simplifying this expression, we get:

(a + 4b)^3 = a^3 + 12a^2b + 48ab^2 + 64b^3.

Therefore, (a + 4b)^3 has 4 terms.

2. [(a + 4b)^3 * (a + 4b)^3]:
Now, let's consider [(a + 4b)^3 * (a + 4b)^3]. Since we already know that (a + 4b)^3 has 4 terms, we can rewrite the expression as:

[(a + 4b)^3 * (a + 4b)^3] = (a^3 + 12a^2b + 48ab^2 + 64b^3) * (a^3 + 12a^2b + 48ab^2 + 64b^3).

To multiply these expressions, we need to apply the distributive property and combine like terms. Since each term in the first expression will be multiplied by each term in the second expression, the resulting expression will have 4 terms * 4 terms = 16 terms.

3. [(a + 4b)^3 * (a + 4b)^3]^2:
Finally, we need to square the expression [(a + 4b)^3 * (a + 4b)^3]. To do this, we multiply the expression by itself:

[(a + 4b)^3 * (a + 4b)^3]^2 = [(a^3 + 12a^2b + 48ab^2 + 64b^3) * (a^3 + 12a^2b + 48ab^2 + 64b^3)]^2.

Since each term in the expression will be multiplied by each term again, the resulting expression will have 16 terms * 16 terms =