Find the product. Simplify your answer. $$ (4 v+1)(2 v-4) $$

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To find the product of the binomials $$(4v + 1)(2v - 4)$$, we'll use the distributive property (also known as the FOIL method: First, Outer, Inner, Last).

1. **First:** Multiply the first terms in each binomial:
   $$
   4v \times 2v = 8v^2
   $$

2. **Outer:** Multiply the outer terms:
   $$
   4v \times (-4) = -16v
   $$

3. **Inner:** Multiply the inner terms:
   $$
   1 \times 2v = 2v
   $$

4. **Last:** Multiply the last terms in each binomial:
   $$
   1 \times (-4) = -4
   $$

Now, combine all these results:
$$
8v^2 - 16v + 2v - 4
$$

Combine like terms ($$-16v + 2v$$):
$$
8v^2 - 14v - 4
$$

So, the simplified product is:

$$
8v^2 - 14v - 4
$$

**Answer:**  
$$8v^{2} -14v -4$$
The product is $$8v^{2} -14v -4$$.

Find the product. Simplify your answer. \[ (4 v+1)(2 v-4) \]

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