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

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UpStudy AI · Accepted Answer

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:

$$
2a^{3} - a^{2} - 1
$$
The product is $$2a^{3} - a^{2} - 1$$.

The Product of a Trinomial and a Binomial

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