Unfortunately, this inequality does not factor easily, so we can try to analyze it by finding the roots of the equation 4x^2 + 5x + 5 = 0 using the quadratic formula:
x = (-B ± sqrt(B^2 - 4AC)) / 2 x = (-5 ± sqrt(5^2 - 445)) / 2* x = (-5 ± sqrt(25 - 80)) / x = (-5 ± sqrt(-55)) / x = (-5 ± i√55) / 8
Since the discriminant is negative, the roots are complex. Therefore, the inequality 4x^2 + 5x + 5 < 0 has no real solutions, and the original inequality does not hold for any real values of x.
Expanding the first two terms:
(4x - 5)(4x + 5) = 16x^2 - 20x + 20x - 25 = 16x^2 - 25
Expanding the last two terms:
-(4x + 5) = -4x - 5
(4x + 1)(3x - 8) = 12x^2 - 32x + 3x - 8 = 12x^2 - 29x - 8
Putting it all together:
(16x^2 - 25) - (4x + 5) - (12x^2 - 29x - 8) < 15x - 2
16x^2 - 25 - 4x - 5 - 12x^2 + 29x + 8 < 15x - 2
4x^2 + 20x - 22 < 15x - 2
4x^2 + 20x - 15x + 27 - 22 <
4x^2 + 5x + 5 < 0
Unfortunately, this inequality does not factor easily, so we can try to analyze it by finding the roots of the equation 4x^2 + 5x + 5 = 0 using the quadratic formula:
x = (-B ± sqrt(B^2 - 4AC)) / 2
x = (-5 ± sqrt(5^2 - 445)) / 2*
x = (-5 ± sqrt(25 - 80)) /
x = (-5 ± sqrt(-55)) /
x = (-5 ± i√55) / 8
Since the discriminant is negative, the roots are complex. Therefore, the inequality 4x^2 + 5x + 5 < 0 has no real solutions, and the original inequality does not hold for any real values of x.