To multiply the polynomials \( (2x - 1)(x^2 + x + 1) \), we can use the distributive property (also known as the FOIL method for binomials). Here, we'll distribute each term in the first polynomial across each term in the second polynomial.
\[ (2x - 1)(x^2 + x + 1) = 2x(x^2) + 2x(x) + 2x(1) - 1(x^2) - 1(x) - 1(1) \]
Now, calculating each of these products:
- \( 2x \cdot x^2 = 2x^3 \)
- \( 2x \cdot x = 2x^2 \)
- \( 2x \cdot 1 = 2x \)
- \( -1 \cdot x^2 = -x^2 \)
- \( -1 \cdot x = -x \)
- \( -1 \cdot 1 = -1 \)
Now combining all these results:
\[ 2x^3 + 2x^2 + 2x - x^2 - x - 1 \]
Next, we combine like terms:
- The \( x^2 \) terms: \( 2x^2 - x^2 = x^2 \)
- The \( x \) terms: \( 2x - x = x \)
So we can rewrite the expression as:
\[ 2x^3 + x^2 + x - 1 \]
Thus, the product of the polynomials \( (2x - 1)(x^2 + x + 1) \) is:
\[ \boxed{2x^3 + x^2 + x - 1} \]