` 3 ( 2x + 1 ) ( 4 x + 1 ) `
Which of the following is equivalent to the expression above?

Options

A. ` 45 x `
B. ` 24 x ^ 2 + 3 `
C. `24 x ^ 2 + 18 x + 3 `
D. ` 18 x ^ 2 + 6 `

Explanation

To solve the expression \( 3(2x + 1)(4x + 1) \), we will follow these steps:

### Step 1: Expand the expression inside the parentheses
We start with the expression \( (2x + 1)(4x + 1) \). We will use the distributive property (also known as the FOIL method for binomials) to expand it.

\[ (2x + 1)(4x + 1) = 2x \cdot 4x + 2x \cdot 1 + 1 \cdot 4x + 1 \cdot 1 \]

### Step 2: Calculate each term
Now we calculate each term:

1. \( 2x \cdot 4x = 8x^2 \)
2. \( 2x \cdot 1 = 2x \)
3. \( 1 \cdot 4x = 4x \)
4. \( 1 \cdot 1 = 1 \)

Putting these together, we have:

\[ 8x^2 + 2x + 4x + 1 \]

### Step 3: Combine like terms
Next, we combine the like terms \( 2x \) and \( 4x \):

\[ 8x^2 + (2x + 4x) + 1 = 8x^2 + 6x + 1 \]

### Step 4: Multiply by 3
Now we multiply the entire expression by 3:

\[ 3(8x^2 + 6x + 1) = 3 \cdot 8x^2 + 3 \cdot 6x + 3 \cdot 1 \]

Calculating each term gives us:

1. \( 3 \cdot 8x^2 = 24x^2 \)
2. \( 3 \cdot 6x = 18x \)
3. \( 3 \cdot 1 = 3 \)

Putting it all together, we have:

\[ 24x^2 + 18x + 3 \]

### Final Result
Thus, the expression \( 3(2x + 1)(4x + 1) \) is equivalent to:

\[ \boxed{24x^2 + 18x + 3} \]