Amanita, let's solve the equation inside the green rectangle step by step. The equation is:

$\frac{2x + 1}{x + 5} = \frac{3x - 1}{x + 7}$

Here’s the plan to solve it:

Step 1: Understand the equation

• We have two fractions set equal to each other. Each fraction has a variable $x$ in both the numerator and the denominator.

Step 2: Find the values of $x$ that make the denominators zero

• The denominators $x + 5$ and $x + 7$ cannot be zero, so $x \neq -5$ and $x \neq -7$.

Step 3: Cross-multiply to eliminate the denominators

• To solve for $x$, we multiply both sides by $(x + 5)(x + 7)$ to get rid of the denominators.

Step 4: Expand and simplify the resulting equation

• After cross-multiplying, we will expand both sides and collect like terms.

Step 5: Solve the resulting linear equation for $x$

• We will isolate $x$ and find its value.

Step 6: Check the solution

• We will make sure the solution does not make any denominator zero.

Let's start solving:

Step 3: Cross-multiply

$(2x + 1)(x + 7) = (3x - 1)(x + 5)$

Step 4: Expand both sides

Let's expand both sides:

Left side: $(2x + 1)(x + 7)$

Right side: $(3x - 1)(x + 5)$

Let's calculate these expansions. Amanita, after expanding both sides, we get:

Left side: $2x^2 + 15x + 7$

Right side: $3x^2 + 14x - 5$

Now, let's move all terms to one side to set the equation to zero:

$2x^2 + 15x + 7 = 3x^2 + 14x - 5$

Subtract $3x^2 + 14x - 5$ from both sides:

$2x^2 + 15x + 7 - (3x^2 + 14x - 5) = 0$

Let's simplify this expression. After simplifying, we get:

$$-x^2 + x + 12 = 0$$

Now, let's solve this quadratic equation for $x$:

$$-x^2 + x + 12 = 0$$

Let's find the values of $x$ that satisfy this equation. Amanita, the solutions to the equation are:

$x = -3 \quad \text{and} \quad x = 4$

Both values do not make the denominators zero, so they are valid solutions.