To solve the equation \((2x + 3)(x + 1) = (3x + 1)(x + 1)\), we can start by expanding both sides of the equation.

First, expand the left-hand side:
\[
(2x + 3)(x + 1) = 2x(x + 1) + 3(x + 1)
\]
\[
= 2x^2 + 2x + 3x + 3
\]
\[
= 2x^2 + 5x + 3
\]

Next, expand the right-hand side:
\[
(3x + 1)(x + 1) = 3x(x + 1) + 1(x + 1)
\]
\[
= 3x^2 + 3x + x + 1
\]
\[
= 3x^2 + 4x + 1
\]

Now, we have the equation:
\[
2x^2 + 5x + 3 = 3x^2 + 4x + 1
\]

Next, move all terms to one side of the equation to set it to zero:
\[
2x^2 + 5x + 3 - 3x^2 - 4x - 1 = 0
\]
\[
(2x^2 - 3x^2) + (5x - 4x) + (3 - 1) = 0
\]
\[
-x^2 + x + 2 = 0
\]

To make it easier to solve, multiply the entire equation by -1:
\[
x^2 - x - 2 = 0
\]

Now, we solve the quadratic equation \(x^2 - x - 2 = 0\). We can factor this equation:
\[
(x - 2)(x + 1) = 0
\]

Setting each factor equal to zero gives us the solutions:
\[
x - 2 = 0 \quad \text{or} \quad x + 1 = 0
\]
\[
x = 2 \quad \text{or} \quad x = -1
\]

So, the solutions to the equation \((2x + 3)(x + 1) = (3x + 1)(x + 1)\) are:
\[
x = 2 \quad \text{and} \quad x = -1
\]