Determine the solution set for the given inequality: $(x+3)(x-2) \leq 0$

Step 1 ：Understand the problem: We are asked to find the solution set for the inequality \((x+3)(x-2) \leq 0\). This means we need to find all values of x that make this inequality true.

Step 2 ：Solve the inequality: To solve this inequality, we first need to find the critical points, which are the values of x that make the expression equal to zero. We set \((x+3)(x-2) = 0\) and solve for x: \(x+3=0\) or \(x-2=0\). This gives us \(x=-3\) or \(x=2\).

Step 3 ：Test the intervals: We have three intervals to test: \((-\infty, -3)\), \((-3, 2)\), and \((2, \infty)\). Choose a test point in each interval and substitute it into the inequality: For \((-\infty, -3)\), choose \(x=-4\): \((-4+3)(-4-2) = 1(-6) = -6\), which is not greater than or equal to 0. For \((-3, 2)\), choose \(x=0\): \((0+3)(0-2) = 3(-2) = -6\), which is not greater than or equal to 0. For \((2, \infty)\), choose \(x=3\): \((3+3)(3-2) = 6(1) = 6\), which is greater than or equal to 0.

Step 4 ：Write the solution: The solution to the inequality is the interval where the inequality is satisfied, which is \((-3, 2)\). However, since the inequality is less than or equal to 0, we also include the points where the expression equals 0, which are \(x=-3\) and \(x=2\). So, the solution set for the inequality \((x+3)(x-2) \leq 0\) is \([-3, 2]\).

Step 5 ：Check the solution: Substitute \(x=-3\) and \(x=2\) into the inequality to check if they satisfy the inequality: For \(x=-3\): \((-3+3)(-3-2) = 0(-5) = 0\), which is equal to 0. For \(x=2\): \((2+3)(2-2) = 5(0) = 0\), which is equal to 0. Both points satisfy the inequality, so the solution is correct. The final answer is \(\boxed{[-3, 2]}\).