The Product of a Trinomial and a Binomial \( (a-1)\left(2 a^{2}+a+1\right) \)

To find the product of the binomial \((a - 1)\) and the trinomial \((2a^{2} + a + 1)\), we'll use the distributive property, also known as the FOIL (First, Outer, Inner, Last) method for multiplying polynomials. ### Step-by-Step Solution: 1. **Distribute each term in the binomial to each term in the trinomial:** \[ (a - 1)(2a^{2} + a + 1) = a \cdot (2a^{2} + a + 1) - 1 \cdot (2a^{2} + a + 1) \] 2. **Multiply \(a\) by each term in the trinomial:** \[ a \cdot 2a^{2} = 2a^{3} \] \[ a \cdot a = a^{2} \] \[ a \cdot 1 = a \] So, \[ a \cdot (2a^{2} + a + 1) = 2a^{3} + a^{2} + a \] 3. **Multiply \(-1\) by each term in the trinomial:** \[ -1 \cdot 2a^{2} = -2a^{2} \] \[ -1 \cdot a = -a \] \[ -1 \cdot 1 = -1 \] So, \[ -1 \cdot (2a^{2} + a + 1) = -2a^{2} - a - 1 \] 4. **Combine the results from steps 2 and 3:** \[ 2a^{3} + a^{2} + a - 2a^{2} - a - 1 \] 5. **Combine like terms:** \[ 2a^{3} + (a^{2} - 2a^{2}) + (a - a) - 1 = 2a^{3} - a^{2} - 1 \] ### Final Answer: \[ 2a^{3} - a^{2} - 1 \]