Để chứng minh M lớn hơn hoặc bằng 1 với mọi giá trị của x, ta sẽ chứng minh rằng đa thức M không có nghiệm thực.

Đầu tiên, ta sẽ đặt đa thức M dưới dạng chuẩn tắc:

M = ax^4 + bx^3 + cx^2 + dx + e

Thay M vào phương trình ban đầu, ta có:

(ax^4 + bx^3 + cx^2 + dx + e) + (2x - x^4 - 5 - 3x^3 - 5x^2) = 2x - 3x^3 + x^4 - 4

Simplifying the equation, we get:

(a - 1)x^4 + (b - 3)x^3 + (c - 5)x^2 + (d + 2)x + (e - 5) = 2x - 4

So we have the following system of equations:

a - 1 = 0
b - 3 = 0
c - 5 = 0
d + 2 = 2
e - 5 = -4

Solving this system of equations, we get:

a = 1
b = 3
c = 5
d = 0
e = 1

Therefore, the polynomial M is:

M = x^4 + 3x^3 + 5x^2 + 1

To prove that M is greater than or equal to 1 for all values of x, we can take the derivative of M and find its minimum value.

Taking the derivative of M, we get:

M' = 4x^3 + 9x^2 + 10x

Setting M' equal to 0 and solving for x, we get:

4x^3 + 9x^2 + 10x = 0

This equation does not have any real solutions, which means that M' is always positive or always negative. Since M' is positive for all values of x, this means that M is always increasing.

Since M is always increasing, and M(0) = 1, this means that M is greater than or equal to 1 for all values of x.

Therefore, we have proved that M is greater than or equal to 1 for all values of x.