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

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

Step-by-Step Plan

1. Understand the Equation: We have two fractions set equal to each other. To solve for $x$, we need to eliminate the denominators.
2. Cross-Multiply: Since the denominators are different, we can cross-multiply to get rid of the fractions. This means multiplying both sides by both denominators.
3. Expand and Simplify: After cross-multiplying, expand both sides and collect like terms.
4. Solve for $x$: Rearrange the equation to isolate $x$ and solve.
5. Check for Restrictions: Make sure the values of $x$ do not make any denominator zero.

Let's begin!

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Step 1: Cross-Multiply

Cross-multiplying gives us:

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

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Step 2: Expand Both Sides

Let's expand both sides:

• Left side: $(2x + 1)(x + 7) = 2x(x + 7) + 1(x + 7) = 2x^2 + 14x + x + 7 = 2x^2 + 15x + 7$
• Right side: $(3x - 1)(x + 5) = 3x(x + 5) - 1(x + 5) = 3x^2 + 15x - x - 5 = 3x^2 + 14x - 5$

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Step 3: Set the Expanded Sides Equal

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

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Step 4: Rearrange to Form a Quadratic Equation

Move all terms to one side:

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

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

Multiply both sides by -1 to make the quadratic coefficient positive:

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

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Step 5: Solve the Quadratic Equation

Now, let's solve:

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

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Step 6: Check for Restrictions

Remember, $x$ cannot be $-5$ or $-7$ because those values would make the denominators zero in the original equation.

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Now, let's solve the quadratic equation for $x$. Amanita, after solving the quadratic equation $x^2 - x - 12 = 0$, we get two possible solutions:

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

Let's check if these values are allowed. The original denominators were $x + 5$ and $x + 7$. If we substitute $x = -3$ or $x = 4$, neither value makes the denominators zero, so both are valid solutions.

Final Answer:

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