The expression you provided, \( x^2 + 3x - 10 \), is a quadratic equation. To analyze it, let's find its roots by factoring, if possible. The general approach involves rewriting the quadratic in the form \( (x - p)(x - q) = 0 \), where \( p \) and \( q \) are the roots of the equation.

### Step 1: Factorize the Quadratic Equation

The given equation is:

\[
x^2 + 3x - 10 = 0
\]

To factorize, we need two numbers that multiply to \(-10\) (the constant term) and add up to \(3\) (the coefficient of \(x\)).

Factors of \(-10\) that add up to \(3\) are \(5\) and \(-2\). Therefore, we can rewrite the equation as:

\[
x^2 + 5x - 2x - 10 = 0
\]

Grouping the terms:

\[
x(x + 5) - 2(x + 5) = 0
\]

Factoring out the common term \((x + 5)\):

\[
(x - 2)(x + 5) = 0
\]

### Step 2: Find the Roots

Setting each factor equal to zero:

\[
x - 2 = 0 \quad \text{or} \quad x + 5 = 0
\]

Solving these gives:

\[
x = 2 \quad \text{or} \quad x = -5
\]

### Conclusion

The roots of the quadratic equation \( x^2 + 3x - 10 = 0 \) are:

\[
\boxed{x = 2 \text{ and } x = -5}
\]

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

### Additional Questions:
1. How do you graph the function \( x^2 + 3x - 10 \)?
2. What is the vertex of the parabola represented by \( x^2 + 3x - 10 \)?
3. Can you find the axis of symmetry for the equation \( x^2 + 3x - 10 \)?
4. How does changing the constant term \(-10\) affect the roots?
5. What would the discriminant of \( x^2 + 3x - 10 \) tell us about the nature of its roots?

### Tip:
Always check if the quadratic can be factored easily by looking for pairs of factors of the constant term that sum to the coefficient of the middle term.

Discover how to solve the quadratic equation x² + 3x - 10 using factoring. This step-by-step solution demonstrates the process of factoring and finding the roots, suitable for students in Grades 8-10. Key concepts include algebra, quadratic equations, and the zero product property.

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