Completely factor the expression.

36x^3- 96x^2y+64xy^2

=4x(9x^2-24xy+16y^2)=4x(3x-4y)^2

To completely factor the expression 36x^3 - 96x^2y + 64xy^2, we can use a method known as factorization by grouping.

Step 1: Find the greatest common factor (GCF) of all the terms in the expression. In this case, the GCF is 4xy.

Step 2: Apply the distributive property by dividing each term in the expression by the GCF.

(36x^3 - 96x^2y + 64xy^2) / (4xy) = 9x^2 - 24xy + 16y^2

Step 3: Notice that the resulting expression is a quadratic trinomial 9x^2 - 24xy + 16y^2, which can be factored.

To factor the quadratic trinomial, we look for two numbers that multiply to give the product of the leading coefficient (9) and the constant term (16), and add up to the coefficient of the middle term (-24).

In this case, the numbers are -3 and -4, because (-3)(-4) = 12, and (-3) + (-4) = -7.

Step 4: Rewrite the quadratic trinomial by splitting the middle term using the two numbers found in the previous step.

9x^2 - 3xy - 21xy + 16y^2

Step 5: Factor by grouping by grouping the terms with a common factor.

(9x^2 - 3xy) - (21xy - 16y^2)

Step 6: Factor out the GCF from each group.

3x(3x - y) - 8y(3x - y)

Step 7: Notice the common binomial factor (3x - y).

(3x - y)(3x - y)

This is the completely factored expression. Therefore, 36x^3 - 96x^2y + 64xy^2 = (3x - y)(3x - y).