2a² + 4a + 5 = 0

We are given: ${\displaystyle 2a²+4a+5=0}$
Solve by completing the square. First, we write the equation in the form ${\displaystyle x²+bx=c}$. Subtract 5 from both sides: ${\displaystyle 2a²+4a=-5}$
Divide both sides by 2: ${\displaystyle a²+2a=-\left(\frac{5}{2}\right)}$
Complete the square by adding ${\displaystyle {\left(\frac{b}{2}\right)}^2}$ on both sides. Here, b=2 so we add ${\displaystyle {\left(\frac{2}{2}\right)}^2=1}$ to both sides.
${\displaystyle a²+2a+1=-\left(\frac{5}{2}\right)+1}$
${\displaystyle (a+1)²=-\left(\frac{3}{2}\right)}$
Take the square root of both sides: ${\displaystyle a+1=\pm \sqrt{-\frac{3}{2}}}$
The square root of a negative number is not defined so the equation has no real solution.