The given expression is 4sin^2(x) * 3cos^2(x) * sin(x/2) * cos(x/2). We need to find the maximum value of this expression.

To simplify the expression, let's break it down into smaller parts and solve step by step.

Step 1: Simplify the expression
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The expression can be simplified using trigonometric identities. We know that sin^2(x) = 1 - cos^2(x) and cos(2x) = 2cos^2(x) - 1.

Let's apply these identities to the expression:

4sin^2(x) * 3cos^2(x) * sin(x/2) * cos(x/2)
= 4(1 - cos^2(x)) * 3cos^2(x) * sin(x/2) * cos(x/2)
= 4(3cos^2(x) - 3cos^4(x)) * sin(x/2) * cos(x/2)
= 12cos^2(x) - 12cos^4(x)) * sin(x/2) * cos(x/2)

Step 2: Apply double angle formula
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We can further simplify the expression by using the double angle formula for sin(2u) = 2sin(u)cos(u).

sin(x/2) * cos(x/2) = 1/2 * sin(x)

Substituting this back into the expression:

12cos^2(x) - 12cos^4(x)) * sin(x/2) * cos(x/2)
= 12cos^2(x) - 12cos^4(x)) * (1/2 * sin(x))
= 6cos^2(x) * sin(x) - 6cos^4(x) * sin(x)
= 6cos^2(x) * sin(x) - 6sin(x) * cos^4(x)

Step 3: Apply product-to-sum formula
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We can now apply the product-to-sum formula for sin(a)sin(b) = 1/2[cos(a-b)-cos(a+b)].

6cos^2(x) * sin(x) - 6sin(x) * cos^4(x)
= 1/2[6cos^2(x) * sin(x) - 6sin(x) * cos^4(x)]
= 1/2[3cos^2(x) * 2sin(x) - 3sin(x) * 2cos^4(x)]
= 3sin(x)[cos^2(x) - cos^4(x)]

Step 4: Simplify the expression further
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We can now simplify the expression even more. Let's substitute cos^2(x) = 1 - sin^2(x) into the expression:

3sin(x)[cos^2(x) - cos^4(x)]
= 3sin(x)[1 - sin^2(x) - (1 - sin^2(x))^2]
= 3sin(x)[1 - sin^2(x) - (1 - 2sin^2(x) + sin^4(x))]
= 3sin(x)[1 - sin^2(x) - 1 + 2sin^2(x) - sin^4(x)]
= 3sin(x)[2sin^