10(x-2)²-5x(2x-1)<-30 inequation

Given the inequality:
$$- 5 x \left(2 x - 1\right) + 10 \left(x - 2\right)^{2} < -30$$
To solve this inequality, we must first solve the corresponding equation:
$$- 5 x \left(2 x - 1\right) + 10 \left(x - 2\right)^{2} = -30$$
Solve:
Given the equation:

10*(x-2)^2-5*x*(2*x-1) = -30

Expand expressions:

40 - 40*x + 10*x^2 - 5*x*(2*x - 1) = -30

40 - 40*x + 10*x^2 - 10*x^2 + 5*x = -30

Reducing, you get:

70 - 35*x = 0

Move free summands (without x)
from left part to right part, we given:
$$- 35 x = -70$$
Divide both parts of the equation by -35

x = -70 / (-35)

We get the answer: x = 2
$$x_{1} = 2$$
$$x_{1} = 2$$
This roots
$$x_{1} = 2$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} < x_{1}$$
For example, let's take the point
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + 2$$
=
$$\frac{19}{10}$$
substitute to the expression
$$- 5 x \left(2 x - 1\right) + 10 \left(x - 2\right)^{2} < -30$$
$$- \frac{5 \cdot 19}{10} \left(-1 + \frac{2 \cdot 19}{10}\right) + 10 \left(-2 + \frac{19}{10}\right)^{2} < -30$$

-53/2 < -30

but

-53/2 > -30

Then
$$x < 2$$
no execute
the solution of our inequality is:
$$x > 2$$

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