You have the following quadratic equation:
x² + 4x + 11 = 0
In order to find the solutions to the previous equation you use the quadratic formula, which is given by:
x = (-b ± √(b² - 4ac))/2a
the general form of a quadratic equation is:
ax² + bx + c = 0
by comparing the previous general form with the given equation, you have:
a = 1
b = 4
c = 11
replace the previous values into the quadratic formula:
x = (-4 ± √(4² - 4(1)(11)))/2(1)
x = (-4 ± √(16 - 44))/2
x = (-4 ± √(-28))/2
the factor √(-28) can be written as √((-1)(28)) = √(2²7(-1)) = 2√7i
i = √(-1)
then, in the quadratic formula you have:
x = (-4 ± 2√7 i)/2
then, the solutions to the given equation are complex numbers, given by:
x1 = -2 + √7 i
x2 = -2 - √7 i