To find the factors of the quadratic expression x² + 4x + 5, we can use the method of factoring or the quadratic formula.
A standard quadratic expression can be factored into the form:
(x + a)(x + b),
where ab = c (the constant term) and a + b = b (the coefficient of x).
In this case, our expression is:
x² + 4x + 5
Here, a + b = 4 and ab = 5. Now, we look for two numbers that satisfy these conditions:
After examining potential pairs of factors of 5, we find that there are no two numbers that add up to 4. Therefore, this polynomial cannot be factored into rational factors.
Instead, we can use the quadratic formula to find the roots of the expression:
x = (-b ± √(b² – 4ac)) / 2a
For our equation, a = 1, b = 4, and c = 5:
Substituting these values, we get:
x = (-4 ± √(4² – 4 * 1 * 5)) / (2 * 1)
That simplifies to:
x = (-4 ± √(16 – 20)) / 2
This results in:
x = (-4 ± √(-4)) / 2
Since the term inside the square root is negative, this will yield complex roots:
x = -2 ± i
Thus, the expression can be factored as:
(x + 2 – i)(x + 2 + i)
In conclusion, the polynomial x² + 4x + 5 does not have real factors but has complex factors given by:
(x + 2 – i) and (x + 2 + i).