We are given the system of equations:

1. $3x^2 + 3xy + y^2 = 909$
2. $3x^2 + xy + 3y^2 = 1287$

We need to solve for $x + y$.

Step 1: Subtract the equations

We subtract equation (1) from equation (2):

$(3x^2 + xy + 3y^2) - (3x^2 + 3xy + y^2) = 1287 - 909$

Simplifying the left-hand side:

$(3x^2 - 3x^2) + (xy - 3xy) + (3y^2 - y^2) = 1287 - 909$

$0 - 2xy + 2y^2 = 378$

$2y^2 - 2xy = 378$

Dividing through by 2:

y^2 - xy = 189 \tag{3}

Step 2: Solve for one variable in terms of the other

We can solve for $y^2 - xy = 189$ from equation (3). Let's isolate $xy$ in this equation:

$y^2 - xy = 189 \quad \text{(equation 3)}$

We still need to work on combining this with the other equation. Let's try checking for factoring methods or solve one