(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:

\[ x = -\frac{3}{2} \]