Given the function $f(x_1, x_2) = 2x_1^2 + 3x_1x_2 + 3x_2^2 + x_1 + 3x_2$, we need to find the gradient of $f$.

The gradient $\nabla f$ of a multivariable function is the vector of partial derivatives of the function with respect to each variable.

Let's calculate the partial derivatives:

1. Partial derivative with respect to $x_1$:

$\frac{\partial f}{\partial x_1} = \frac{\partial}{\partial x_1} \left( 2x_1^2 + 3x_1x_2 + 3x_2^2 + x_1 + 3x_2 \right)$ $\frac{\partial f}{\partial x_1} = 4x_1 + 3x_2 + 1$

2. Partial derivative with respect to $x_2$:

$\frac{\partial f}{\partial x_2} = \frac{\partial}{\partial x_2} \left( 2x_1^2 + 3x_1x_2 + 3x_2^2 + x_1 + 3x_2 \right)$ $\frac{\partial f}{\partial x_2} = 3x_1 + 6x_2 + 3$

Thus, the gradient of $f$ is: $\nabla f = \left[ 4x_1 + 3x_2 + 1, 3x_1 + 6x_2 + 3 \right]$

From the given options, the correct gradient is: $\boxed{\nabla f = [4x_1 + 3x_2 + 1, 3x_1 + 6x_2 + 3]}$

Do you want more details on this? Here are some related questions:

1. What is the geometric interpretation of the gradient of a function?
2. How do we interpret partial derivatives in the context of multivariable functions?
3. Can the gradient of a function be zero, and what does it signify?
4. How do we find the directional derivative using the gradient?
5. What is the role of the gradient in optimization problems?
6. How is the Hessian matrix related to the gradient?
7. How does the gradient affect the level curves of a function?
8. How do we extend the concept of a gradient to more than two variables?

Tip: The gradient always points in the direction of the steepest ascent of the function.