Is there any real value of 'a' for which the equation x2 + 2x + (a2 + 1) = 0

Let quadratic equation $x^{2}+2 x+\left(a^{2}+1\right)=0$ has real roots.

Here, $a=1, b=2$ and,$c=\left(a^{2}+1\right)$

As we know that $D=b^{2}-4 a c$

Putting the value of $a=1, b=2$ and, $c=\left(a^{2}+1\right)$, we get

$D=(2)^{2}-4 \times 1 \times\left(a^{2}+1\right)$

$=4-4\left(a^{2}+1\right)$

$=-4 a^{2}$

The given equation will have equal roots, if $D>0$

i.e. $-4 a^{2}>0$

$\Rightarrow a^{2}<0$

which is not possible, as the square of any number is always positive.

Thus, No, there is no any real value of $a$ for which the given equation has real roots.