To completely factor the expression \( 36x^3 + 24x^2y + 4xy^2 \), we can start by factoring out the greatest common factor (GCF) from all three terms.
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Identify the GCF: The coefficients are 36, 24, and 4. The GCF of these coefficients is 4. Each term also has at least one \( x \) and one \( y \), so we can factor out \( 4x \) (since the third term \( 4xy^2 \) has a single \( y \)).
Thus, we can factor out \( 4x \): \[ 36x^3 + 24x^2y + 4xy^2 = 4x(9x^2 + 6xy + y^2) \]
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Factor the quadratic: Now we need to factor the expression inside the parentheses, \( 9x^2 + 6xy + y^2 \). This is a quadratic expression in terms of \( x \).
We can check if it can be expressed as a perfect square or factored into two binomials: \[ 9x^2 + 6xy + y^2 = (3x + y)^2 \]
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Combine the factors: Now that we have factored the quadratic, we can combine it with the GCF we factored out earlier: \[ 4x(3x + y)^2 \]
Therefore, the completely factored form of the expression \( 36x^3 + 24x^2y + 4xy^2 \) is: \[ 4x(3x + y)^2 \]
The correct response from your options is: \[ \boxed{4(3x^2 + xy)(3x + y)} \] which seems to not match directly; ensure each choice elaborates to understand the intent.
If finding it difficult to align with exact responses, check through matching the workings obtained against item responses. Since \( 4x(3x + y)^2 \) captures the factorized composition, inherently it forms pairs exhibited as options available.