The expression you've provided is a quadratic expression in the form of $ax^2 + bx + c$, where:

• $a = 1$ (coefficient of $x^2$),
• $b = 4$ (coefficient of $x$),
• $c = 2$ (constant term).

Factoring the Quadratic Expression

This quadratic expression doesn't factor neatly with integer coefficients, so we'll use the quadratic formula to find the roots:

The quadratic formula is:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Substituting the values $a = 1$, $b = 4$, and $c = 2$ into the formula:

$x = \frac{-4 \pm \sqrt{4^2 - 4(1)(2)}}{2(1)}$

$x = \frac{-4 \pm \sqrt{16 - 8}}{2}$

$x = \frac{-4 \pm \sqrt{8}}{2}$

$x = \frac{-4 \pm 2\sqrt{2}}{2}$

$x = -2 \pm \sqrt{2}$

So the roots of the quadratic equation are:

$x = -2 + \sqrt{2} \quad \text{and} \quad x = -2 - \sqrt{2}$

Completing the Square

Another way to express the quadratic is by completing the square:

$x^2 + 4x + 2 = \left(x^2 + 4x + 4\right) - 4 + 2 = (x + 2)^2 - 2$

So, the expression can also be written as:

$(x + 2)^2 - 2$

Would you like further details or have any questions?

Here are some related questions you might find interesting:

1. How do you derive the quadratic formula?
2. What are the applications of completing the square?
3. How does the discriminant help in determining the nature of the roots?
4. Can every quadratic expression be factored over the real numbers?
5. What are the connections between the roots and the graph of the quadratic function?

Tip: When using the quadratic formula, always check the discriminant ($b^2 - 4ac$) first to determine the nature of the roots (real, equal, or complex).