**Solution:**

To simplify the given expression **3(sin theta - cos theta )⁴ + 6(sin theta * cos theta )² + 4sin^6theta**, we can use the following steps:

**Step 1: Expand the expression**

Expand the given expression using the binomial theorem and the trigonometric identities.

The binomial theorem 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 binomial coefficient, which is equal to n! / (r! * (n-r)!).

Applying the binomial theorem to our expression, we have:

3(sin theta - cos theta )⁴ = 3 * (sin^4 theta - 4sin^3 theta * cos theta + 6sin^2 theta * cos^2 theta - 4sin theta * cos^3 theta + cos^4 theta)

6(sin theta * cos theta )² = 6 * (sin^2 theta * cos^2 theta)

4sin^6theta = 4 * (sin^6 theta)

**Step 2: Combine like terms**

Now, let's combine the like terms in our expanded expression:

3 * (sin^4 theta - 4sin^3 theta * cos theta + 6sin^2 theta * cos^2 theta - 4sin theta * cos^3 theta + cos^4 theta) + 6 * (sin^2 theta * cos^2 theta) + 4 * (sin^6 theta)

= 3sin^4 theta - 12sin^3 theta * cos theta + 18sin^2 theta * cos^2 theta - 12sin theta * cos^3 theta + 3cos^4 theta + 6sin^2 theta * cos^2 theta + 4sin^6 theta

**Step 3: Simplify further**

Now, let's simplify the expression further by combining the terms with similar powers of sin and cos:

3sin^4 theta - 12sin^3 theta * cos theta + 18sin^2 theta * cos^2 theta - 12sin theta * cos^3 theta + 3cos^4 theta + 6sin^2 theta * cos^2 theta + 4sin^6 theta

= 4sin^6 theta + 3sin^4 theta - 12sin^3 theta * cos theta + 24sin^2 theta * cos^2 theta - 12sin theta * cos^3 theta + 3cos^4 theta

Now, we can see that the expression is simplified to:

**4sin^6 theta + 3sin^4 theta - 12sin^3 theta * cos theta + 24sin^2 theta * cos^2 theta - 12sin theta * cos^3 theta + 3cos^4 theta**

Therefore, the simplified form of the given expression is **4sin^6 theta + 3sin^4 theta - 12sin^3 theta * cos theta + 24sin^2 theta * cos^2 theta - 12sin theta * cos^3 theta