1. Factor the leading coefficient out of the first two terms

$a x^2\+b x\+c \=$

$a\left(x^2 \+\frac{b}{a}x \right)\+c$

2. Complete the square by adding and subtracting the "magic number"

(the square of half the coefficient of $x$)

$\=$

$a\left(x^2\+ \frac{b}{a}x \+ {\left(\frac{b}{2 a}\right)}^2 - {\left(\frac{b}{2 a}\right)}^2 \right)\+c$

3. Move the constant $-{\left(\frac{b}{2 a}\right)}^2$ outside the parentheses. Remember to multiply it by a.

$\=$

$a\left(x^2\+ \frac{b}{a}x \+ {\left(\frac{b}{2 a}\right)}^2 \right)\+c - \frac{b^2}{4 a}$

4. Factor the perfect square and add the remaining terms.

$\=$

$a{\left(x^{}\+ \frac{b}{2 a} \right)}^2\+c - \frac{b^2}{4 a}$

1. Factor the leading coefficient out of the first two terms

$3 x^2\+6 x\+4 \=$

$3\left(x^2 \+2 x \right)\+4$

2. Complete the square by adding and subtracting the "magic number"

(the square of half the coefficient of x)

$\=$

$3\left(x^2\+ 2 x \+ 1 - 1 \right)\+4$

3. Move the constant $-1$ outside the parentheses. Remember to multiply it by 3.

$\=$

$3\left(x^2\+ 2 x \+1\right)\+4 -3$

4. Factor the perfect square and add the remaining terms.

$\=$

$3{\left(x^{}\+ 1\right)}^2\+1$