To solve the equation \( \frac{2}{x} + 4 = x + 3 \), we need to isolate \( x \). Here's the step-by-step process:

### Step 1: Subtract 3 from both sides
\[
\frac{2}{x} + 4 - 3 = x + 3 - 3
\]
This simplifies to:
\[
\frac{2}{x} + 1 = x
\]

### Step 2: Subtract 1 from both sides
\[
\frac{2}{x} + 1 - 1 = x - 1
\]
This simplifies to:
\[
\frac{2}{x} = x - 1
\]

### Step 3: Multiply both sides by \( x \) to eliminate the denominator
\[
x \cdot \frac{2}{x} = (x - 1) \cdot x
\]
This simplifies to:
\[
2 = x(x - 1)
\]
Which expands to:
\[
2 = x^2 - x
\]

### Step 4: Rearrange the equation
\[
x^2 - x - 2 = 0
\]

### Step 5: Factor the quadratic equation
\[
x^2 - x - 2 = (x - 2)(x + 1) = 0
\]

### Step 6: Solve for \( x \)
From \( (x - 2)(x + 1) = 0 \), we get two solutions:
\[
x - 2 = 0 \quad \text{or} \quad x + 1 = 0
\]
\[
x = 2 \quad \text{or} \quad x = -1
\]

### Step 7: Check for extraneous solutions
Since the original equation contains \( \frac{2}{x} \), \( x = 0 \) is not a valid solution. However, \( x = 2 \) and \( x = -1 \) are valid.

Thus, the solutions are:
\[
x = 2 \quad \text{or} \quad x = -1
\]

Would you like more details on any part of the solution, or do you have any questions?

Here are 5 related questions:
1. How do we solve equations with fractions and variables in the denominator?
2. What is the method to factor quadratic equations?
3. What is an extraneous solution, and why do we check for it?
4. How do you complete the square to solve quadratic equations?
5. How can we solve quadratic equations using the quadratic formula?

**Tip:** Always check for extraneous solutions when dealing with fractions in an equation to avoid invalid answers.

Learn how to solve the equation 2/x + 4 = x + 3 with a step-by-step explanation. This problem involves solving rational equations and factoring quadratic equations. Suitable for students in Grades 9-11.

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