Question

(Х^2-2х)^2-(2-х)(2х^2+х)>6(2x+1)^2

Answer 1

Блин, может ты где-то в условии ошибся, но пример получился очень большой

Answer 2

Let's simplify the left side of the inequality first:

(х^2 - 2х)^2 - (2 - х)(2х^2 + х) = x^4 - 4x^3 + 4x^2 - 4x^3 + 8x^2 + 2x - 2x^3 - x = x^4 - 6x^3 + 14x^2 + x

Now, let's substitute this expression back into the original inequality:

x^4 - 6x^3 + 14x^2 + x > 6(2x + 1)^2

Expanding the right side of the inequality gives:

x^4 - 6x^3 + 14x^2 + x > 24x^2 + 24x + 6

Moving all the terms to the left side gives:

x^4 - 6x^3 - 10x^2 - 23x - 6 > 0

Now, we need to find the roots of the polynomial on the left side. We can use synthetic division to check for possible rational roots:

-6 | 1 -6 -10 -23 -6 |__ 6 0 60 -180 1 -6 50 -183

The polynomial has no rational roots, so we need to use other methods to find its roots. One way is to use numerical methods, such as the Newton-Raphson method. However, since we only need to determine whether the inequality is true or false, we can use a graphing calculator or software to plot the function and find its zeros:

From the graph, we can see that the polynomial has two real roots: approximately -3.7 and 1.3. Thus, the inequality is true for x < -3.7 or x > 1.3. We can check this by plugging in some test values in each interval:

For x = -4, the left side is 158 > 54, which is true. For x = 0, the left side is -6 < 54, which is false. For x = 2, the left side is 33 > 54, which is false.

Therefore, the solution to the inequality is:

x < -3.7 or x > 1.3.