First divide both sides by 2:
[tex]2x^2+7x+6=0\implies x^2+\dfrac72x+3=0[/tex]
Move the constant term to the right side:
[tex]x^2+\dfrac72x=-3[/tex]
Since
[tex](x+y)^2=x^2+2xy+y^2[/tex]
we want to have
[tex]2y=\dfrac72\implies y=\dfrac74[/tex]
Then
[tex]\left(x+\dfrac74\right)^2=x^2+\dfrac72x+\dfrac{49}{16}[/tex]
So in our equation, we need to add 49/16 to both sides in order to complete the square:
[tex]x^2+\dfrac72x+\dfrac{49}{16}=-3+\dfrac{49}{16}[/tex]
[tex]\implies\left(x+\dfrac74\right)^2=\dfrac1{16}[/tex]
