To determine the product of -4 and [tex]\(3 \frac{1}{4}\)[/tex], we first need to understand how to handle the mixed number [tex]\(3 \frac{1}{4}\)[/tex].
A mixed number [tex]\(3 \frac{1}{4}\)[/tex] can be converted to an improper fraction:
[tex]\[ 3 \frac{1}{4} = 3 + \frac{1}{4} = \frac{12}{4} + \frac{1}{4} = \frac{13}{4} \][/tex]
Therefore, the task becomes finding the product of -4 and [tex]\(\frac{13}{4}\)[/tex].
We split this product into two parts:
1. The integer part: [tex]\(3\)[/tex]
2. The fractional part: [tex]\(\frac{1}{4}\)[/tex]
We can express the product [tex]\(-4 \cdot 3 \frac{1}{4}\)[/tex] as:
[tex]\[ (-4) \cdot \left(3 + \frac{1}{4}\right) \][/tex]
Using the distributive property:
[tex]\[ (-4) \cdot 3 + (-4) \cdot \frac{1}{4} \][/tex]
So, we can rewrite this as:
[tex]\[ (-4)(3) + (-4)\left(\frac{1}{4}\right) \][/tex]
Now, let's look at our options:
1. [tex]\((-4)(3) \times (-4)\left( \frac{1}{4}\right) \)[/tex]
2. [tex]\((-4)(3) + (-4)\left( \frac{1}{4}\right) \)[/tex]
3. [tex]\((-4) \times (3) \left(\frac{1}{4}\right) \)[/tex]
4. [tex]\((3)(-4) + (3) \left( \frac{1}{4}\right) \)[/tex]
Of the listed options, the correct expression that would determine the product of [tex]\(-4\)[/tex] and [tex]\(3 \frac{1}{4}\)[/tex] is:
[tex]\[ (-4)(3) + (-4)\left(\frac{1}{4}\right) \][/tex]
Thus, the correct answer is:
[tex]\[
\boxed{(-4)(3) + (-4)\left(\frac{1}{4}\right)}
\][/tex]
