Calculate: (4x+2)^2+4(4x+2)+4=0

Expression: ${\left( 4x+2 \right)}^{2}+4\left( 4x+2 \right)+4=0$

Use ${\left( a+b \right)}^{2}={a}^{2}+2ab+{b}^{2}$ to expand the expression

$16{x}^{2}+16x+4+4\left( 4x+2 \right)+4=0$

Distribute $4$ through the parentheses

$16{x}^{2}+16x+4+16x+8+4=0$

Collect like terms

$16{x}^{2}+32x+4+8+4=0$

Calculate the sum of the positive numbers

$16{x}^{2}+32x+16=0$

Divide both sides of the equation by $16$

${x}^{2}+2x+1=0$

Identify the coefficients $a$, $b$ and $c$ of the quadratic equation

$\begin{array} { l }a=1,& b=2,& c=1\end{array}$

Substitute $a=1$, $b=2$ and $c=1$ into the quadratic formula $x=\frac{ -b\pm\sqrt{ {b}^{2}-4ac } }{ 2a }$

$x=\frac{ -2\pm\sqrt{ {2}^{2}-4 \times 1 \times 1 } }{ 2 \times 1 }$

Any expression multiplied by $1$ remains the same

$x=\frac{ -2\pm\sqrt{ {2}^{2}-4 \times 1 } }{ 2 \times 1 }$

Any expression multiplied by $1$ remains the same

$x=\frac{ -2\pm\sqrt{ {2}^{2}-4 } }{ 2 \times 1 }$

Any expression multiplied by $1$ remains the same

$x=\frac{ -2\pm\sqrt{ {2}^{2}-4 } }{ 2 }$

Evaluate the power

$x=\frac{ -2\pm\sqrt{ 4-4 } }{ 2 }$

The sum of two opposites equals $0$

$x=\frac{ -2\pm\sqrt{ 0 } }{ 2 }$

Any root of $0$ equals $0$

$x=\frac{ -2\pm0 }{ 2 }$

Removing $0$ doesn't change the value, so remove it from the expression

$x=\frac{ -2 }{ 2 }$

Any expression divided by its opposite equals $-1$

$x=-1$