This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Step 1: Analyze the numerator of the inequality. The given inequality is ((x+1)^2+3)/((x+3)(x-2)) < 0. Consider the numerator: (x+1)^2+3. Since (x+1)^2 is always greater than or equal to 0 for any real number x, adding 3 to it means that (x+1)^2+3 is always greater than or equal to 3. Therefore, the numerator (x+1)^2+3 is always positive. Step 2: Determine the sign of the denominator. For the entire fraction to be less than 0 (negative), and since the numerator is always positive, the denominator must be negative. So, we must have (x+3)(x-2) < 0. Step 3: Solve the inequality for the denominator. To find when (x+3)(x-2) < 0, we find the roots of the quadratic expression (x+3)(x-2) = 0. The roots are x = -3 and x = 2. This is a parabola opening upwards. The expression (x+3)(x-2) will be negative when x is between its roots. Thus, the solution to (x+3)(x-2) < 0 is -3 < x < 2. Step 4: Compare with the given options. The set of values for x is \x : -3 < x < 2\. This matches option D. The final answer is D.