To factor the given expression [tex]\( x^2 + 11 \)[/tex] into its complex factors, we use the concept of the difference of squares and the fact that [tex]\( -1 \)[/tex] can introduce complex units.
Recall that:
[tex]\[ x^2 + a^2 = (x + ai)(x - ai) \][/tex]
where [tex]\( i \)[/tex] is the imaginary unit and [tex]\( a \)[/tex] is a positive real number. In this case, [tex]\( a^2 = 11 \)[/tex], so [tex]\( a = \sqrt{11} \)[/tex].
Thus, we can express [tex]\( x^2 + 11 \)[/tex] as:
[tex]\[ x^2 + (\sqrt{11})^2 \][/tex]
Using the above factorization formula, we get:
[tex]\[ x^2 + 11 = (x + \sqrt{11}i)(x - \sqrt{11}i) \][/tex]
Therefore, the correct factorization of [tex]\( x^2 + 11 \)[/tex] into its complex factors is:
[tex]\[ (x + \sqrt{11}i)(x - \sqrt{11}i) \][/tex]
So, the correct answer is:
D. [tex]\( (x + i\sqrt{11})(x - i\sqrt{11}) \)[/tex]
