Take a look at the expression below:

x^{2 }+ 4x + 3

To complete the square you have to focus on the number before the x or the (x coefficient).

In this case, this number is 4.

To complete the square you take that number (x coefficent) and halve it, then square it.

Therefore: 4/2 = 2 -----> 2^{2 }= 4

We then add this number after the x and also minus it after the last number (constant):

x^{2 }+ 4x + 4 + 3 - 4

Completing the square is about being able to factorise, which is why this expression can now be factorised:

(x^{2} + 4x + 4) + 3 - 4

The brackets factorise to --> (x + 2)^{2} whilst the digits outside the brackets equate to -1

Therefore, our completed expression would now look like: (x + 2)^{2} - 1

The reason this is useful is because if our original expression was an equation it would look like this:

x^{2 }+ 4x + 3 = 0

Therefore, our new equation would look like this:

(x+2)^{2} - 1 = 0

With our original equation the only way we could solve it is by using the quadratic formula.

But with our new factorised equation we can solve for x by quick algebra manipulation:

--> (x+2)^{2} - 1 = 0

--> (x+2)^{2} = 1

--> (x+2) = sqrt(1)

--> x + 2 = 1

--> x = 1 - 2

Therefore: x = -1