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To solve this problem, we need to analyze the number line and determine which quadratic inequality it represents.

### Step 1: Observing the Number Line
The number line shows the solution region between -4 and 6, where the region is shaded or included (solid red line), indicating that the values between -4 and 6 are part of the solution. The solid endpoints suggest the inequality includes the values -4 and 6 (≤ or ≥).

### Step 2: Identifying the Quadratic Expression
The two relevant factors for this inequality are the roots of the quadratic equation, which appear to be \( x = -4 \) and \( x = 6 \). Therefore, the factors of the quadratic expression can be written as:
\[
(x + 4)(x - 6)
\]

### Step 3: Determining the Correct Inequality
Since the region between the roots is shaded, we look for where the expression is less than or equal to zero:
\[
(x + 4)(x - 6) \leq 0
\]
This inequality matches the shaded region on the number line, where the quadratic expression is non-positive.

Thus, the correct answer is:
\[
(x + 4)(x - 6) \leq 0
\]

Would you like more details or have any questions?

Here are some related questions:
1. How do you solve quadratic inequalities in general?
2. What is the significance of the roots in a quadratic inequality?
3. How does the sign of the product affect the solution set of a quadratic inequality?
4. How can you determine if the endpoints should be included in the solution?
5. How do you graphically represent the solutions to inequalities?

**Tip:** When solving quadratic inequalities, always determine the intervals where the expression is positive or negative by testing points in each interval defined by the roots.

The solution of which quadratic inequality is shown on the number line?

Learn how to solve the quadratic inequality (x + 4)(x - 6) ≤ 0 by analyzing its roots and solution on a number line. This step-by-step guide covers key algebraic concepts like factoring and determining intervals of positivity and negativity. Suitable for students in Grades 8-10.

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