To solve the quadratic equation \(4x^2 + 7x + 8 = 6x + x^2 + 10\), follow these steps:

1. **Move all terms to one side** to set the equation to zero:
   \[
   4x^2 + 7x + 8 - 6x - x^2 - 10 = 0
   \]

2. **Simplify** the equation by combining like terms:
   \[
   (4x^2 - x^2) + (7x - 6x) + (8 - 10) = 0
   \]
   \[
   3x^2 + x - 2 = 0
   \]

3. **Solve the quadratic equation** \(3x^2 + x - 2 = 0\) using the quadratic formula:
   \[
   x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
   \]
   where \(a = 3\), \(b = 1\), and \(c = -2\).

4. **Substitute the values**:
   \[
   x = \frac{-1 \pm \sqrt{1^2 - 4(3)(-2)}}{2(3)}
   \]
   \[
   x = \frac{-1 \pm \sqrt{1 + 24}}{6}
   \]
   \[
   x = \frac{-1 \pm \sqrt{25}}{6}
   \]
   \[
   x = \frac{-1 \pm 5}{6}
   \]

5. **Calculate the two possible solutions**:
   - \(x = \frac{-1 + 5}{6} = \frac{4}{6} = \frac{2}{3}\)
   - \(x = \frac{-1 - 5}{6} = \frac{-6}{6} = -1\)

The solutions are \(x = \frac{2}{3}\) and \(x = -1\).

Would you like more details, or do you have any questions?

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Here are 5 related questions you might find interesting:

1. How do you derive the quadratic formula?
2. Can you solve a quadratic equation by factoring instead of using the quadratic formula?
3. What is the significance of the discriminant in a quadratic equation?
4. How do you complete the square to solve a quadratic equation?
5. How does the graph of a quadratic equation relate to its solutions?

**Tip:** Always check your solutions by substituting them back into the original equation to ensure they satisfy the equation.

Learn how to solve the quadratic equation 4x^2 + 7x + 8 = 6x + x^2 + 10 with detailed steps and explanations. This problem involves algebraic manipulation and the application of the quadratic formula. Suitable for students in Grades 10-12.

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