## Exploring the Quadratic Expression (x^2 + x + 1)^2

The expression (x^2 + x + 1)^2 is a fascinating quadratic expression that can be explored through various mathematical methods. Let's delve into its properties and analyze its behavior:

### Expanding the Expression

Firstly, we can expand the expression using the FOIL (First, Outer, Inner, Last) method:

(x^2 + x + 1)^2 = (x^2 + x + 1)(x^2 + x + 1)

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

= **x^4 + 2x^3 + 3x^2 + 2x + 1**

### Factorization and Roots

The expanded expression can be factored by grouping:

x^4 + 2x^3 + 3x^2 + 2x + 1 = (x^4 + x^3 + x^2) + (x^3 + x^2 + x) + (x^2 + x + 1)

= x^2(x^2 + x + 1) + x(x^2 + x + 1) + (x^2 + x + 1)

= **(x^2 + x + 1)(x^2 + x + 1)**

Therefore, the expression has **two identical factors**: (x^2 + x + 1).

To find the roots of the expression, we set it equal to zero:

(x^2 + x + 1)^2 = 0

This implies that (x^2 + x + 1) = 0. This quadratic equation has no real roots as its discriminant (b^2 - 4ac) is negative. However, it has **two complex roots**.

### Analyzing the Expression's Behavior

The expression (x^2 + x + 1)^2 is always **positive or zero** for all real values of x. This is because the expression (x^2 + x + 1) is always positive, as its minimum value is 3/4. Therefore, squaring a positive value always results in a positive value.

### Graphing the Expression

The graph of the expression y = (x^2 + x + 1)^2 is a **symmetrical curve** that is always above the x-axis. It has a **minimum point** at (0,1). The curve rises rapidly as x increases or decreases, demonstrating the effect of squaring the expression.

### Conclusion

The expression (x^2 + x + 1)^2 offers a unique mathematical exploration. Its expansion, factorization, lack of real roots, and always positive behavior highlight the properties of quadratic expressions and their interplay with complex numbers.