(4х2+12х+9)(3х-7х2-18)>=0

To solve the inequality \((4x^2 + 12x + 9)(3x - 7x^2 - 18) \geq 0\), we first need to find the roots of each factor and determine where the product is non-negative. ### Step 1: Factor or find roots of \(4x^2 + 12x + 9\) We can start by factoring \(4x^2 + 12x + 9\): \[ 4x^2 + 12x + 9 = (2x + 3)^2 \] This quadratic has a double root at: \[ 2x + 3 = 0 \quad \Rightarrow \quad x = -\frac{3}{2} \] ### Step 2: Find roots of \(3x - 7x^2 - 18\) Now we can rearrange \(3x - 7x^2 - 18\): \[ -7x^2 + 3x - 18 = 0 \] Multiplying through by \(-1\) gives: \[ 7x^2 - 3x + 18 = 0 \] We'll use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\): \[ a = 7, \quad b = -3, \quad c = 18 \] Calculating the discriminant: \[ D = (-3)^2 - 4(7)(18) = 9 - 504 = -495 \] Since \(D < 0\), the second quadratic \(3x - 7x^2 - 18\) has no real roots, indicating it's always positive or always negative. ### Step 3: Determine the sign of \(3x - 7x^2 - 18\) To determine whether \(3x - 7x^2 - 18\) is positive or negative, we can evaluate it at a specific point. Let's test \(x = 0\): \[ 3(0) - 7(0^2) - 18 = -18 < 0 \] Since the parabola opens downward (as the coefficient of \(x^2\) is negative), this quadratic is always negative: \[ 3x - 7x^2 - 18 < 0 \quad \forall x \] ### Step 4: Analyze the original expression The original expression now can be analyzed: \[(2x + 3)^2 < 0 \quad \text{and} \quad (3x - 7x^2 - 18) < 0\] The expression \((2x + 3)^2\) is always non-negative (it is equal to zero at \(x = -\frac{3}{2}\) and positive elsewhere). However, the other factor is always negative. ### Step 5: Conclusion The product \((2x + 3)^2 (3x - 7x^2 - 18)\) can only be zero or negative: \[ (2x + 3)^2 \geq 0 \] \[ 3x - 7x^2 - 18 < 0 \] Thus, the expression \((4x^2 + 12x + 9)(3x - 7x^2 - 18) \geq 0\) holds true only when: \[ (2x + 3)^2 = 0 \quad \Rightarrow \quad x = -\frac{3}{2} \] Therefore, the solution to the inequality \((4x^2 + 12x + 9)(3x - 7x^2 - 18) \geq 0\) is: \[ x = -\frac{3}{2} \]