To multiply the polynomials \( (2x + y)(4x^2 - 2xy + y^2) \), we will use the distributive property (also known as the FOIL method for binomials).
We will distribute each term in the first polynomial to each term in the second polynomial:
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Multiply \( 2x \) by each term in \( (4x^2 - 2xy + y^2) \):
- \( 2x \cdot 4x^2 = 8x^3 \)
- \( 2x \cdot (-2xy) = -4x^2y \)
- \( 2x \cdot y^2 = 2xy^2 \)
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Multiply \( y \) by each term in \( (4x^2 - 2xy + y^2) \):
- \( y \cdot 4x^2 = 4x^2y \)
- \( y \cdot (-2xy) = -2xy^2 \)
- \( y \cdot y^2 = y^3 \)
Now we combine all the results:
\[ 8x^3 - 4x^2y + 2xy^2 + 4x^2y - 2xy^2 + y^3 \]
Next, we combine like terms:
- The \( x^2y \) terms: \( -4x^2y + 4x^2y = 0 \) (these cancel out)
- The \( xy^2 \) terms: \( 2xy^2 - 2xy^2 = 0 \) (these also cancel out)
So, we are left with:
\[ 8x^3 + y^3 \]
Therefore, the product of the polynomials \( (2x + y)(4x^2 - 2xy + y^2) \) is:
\[ \boxed{8x^3 + y^3} \]
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