If `g(x)=(2x+3)^(2)`, then `g((x)/(3)-1)=`

Options

A. `(4x^(2))/(9)+(4x)/(3)+4`
B. `(4x^(2))/(3)+2x+2`
C. `(4x^(2)+2x)/(3)+1`
D. `(4x^(2))/(9)+(4x)/(3)+1`

Explanation

To solve the problem, we need to evaluate \( g\left(\frac{x}{3} - 1\right) \) given that \( g(x) = (2x + 3)^2 \).

### Step-by-Step Solution:

1. **Substitute \( \frac{x}{3} - 1 \) into \( g(x) \)**:
\[ g\left(\frac{x}{3} - 1\right) = \left(2\left(\frac{x}{3} - 1\right) + 3\right)^2 \]

2. **Simplify the expression inside the parentheses**:
\[ 2\left(\frac{x}{3} - 1\right) + 3 = \frac{2x}{3} - 2 + 3 \]
\[ = \frac{2x}{3} + 1 \]

3. **Now substitute this back into the equation**:
\[ g\left(\frac{x}{3} - 1\right) = \left(\frac{2x}{3} + 1\right)^2 \]

4. **Expand the squared term**:
\[ = \left(\frac{2x}{3}\right)^2 + 2 \cdot \frac{2x}{3} \cdot 1 + 1^2 \]
\[ = \frac{4x^2}{9} + \frac{4x}{3} + 1 \]

5. **Combine the terms**:
\[ g\left(\frac{x}{3} - 1\right) = \frac{4x^2}{9} + \frac{4x}{3} + 1 \]

### Final Answer:
\[ g\left(\frac{x}{3} - 1\right) = \frac{4x^2}{9} + \frac{4x}{3} + 1 \]