We are given two equations: $x + y = 1$ and $x + 3y = 5$. We need to find the value of the expression $x^2 + 4xy + 3y^2$.

1. Problem Description

We are given two equations: $x + y = 1$ and $x + 3y = 5$. We need to find the value of the expression $x^2 + 4xy + 3y^2$.

2. Solution Steps

First, we solve the system of equations for $x$ and $y$.

From the first equation, $x = 1 - y$.

Substitute this into the second equation:

$(1 - y) + 3y = 5$

$1 + 2y = 5$

$2y = 4$

$y = 2$

Now, substitute $y = 2$ back into the equation $x = 1 - y$:

$x = 1 - 2$

$x = -1$

Now that we have the values of $x$ and $y$, we can evaluate the expression $x^2 + 4xy + 3y^2$:

$x^2 + 4xy + 3y^2 = (-1)^2 + 4(-1)(2) + 3(2)^2$

$= 1 - 8 + 3(4)$

$= 1 - 8 + 12$

$= 13 - 8$

$= 5$

3. Final Answer

The value of $x^2 + 4xy + 3y^2$ is

5.