To factor the quadratic expression x² + 2x + 8, we first need to analyze the coefficients and the constant term. In this case, the expression is in the standard form of a quadratic, which is:
ax² + bx + c
Here, a = 1, b = 2, and c = 8.
Next, we look for two numbers that multiply to ac (which is 1 * 8 = 8) and add to b (which is 2). However, upon examining the possible factors of 8 (like 1, 2, 4, 8), we find that no two numbers satisfy both conditions.
Thus, this quadratic does not factor neatly into real numbers, indicating it does not have real roots. Instead, we can complete the square or use the quadratic formula to find the roots, but in terms of factoring over the reals, we conclude:
The expression x² + 2x + 8 cannot be factored into real-number factors.
If you want to explore complex roots, you can use the quadratic formula:
x = (-b ± √(b² – 4ac)) / 2a
Substituting the values:
x = (-2 ± √(2² – 4 * 1 * 8)) / (2 * 1) = (-2 ± √(-28)) / 2
So, the roots are:
x = -1 ± √7i
Thus, if you want to express the quadratic as a product of complex factors, it can be represented as:
(x + 1 – √7i)(x + 1 + √7i)