Solve for: (x^2+8x+12)/(x^2+12x+36) * (x^2+6x)/(x^2+(4x)+4)

Expression: $\frac{ {x}^{2}+8x+12 }{ {x}^{2}+12x+36 } \times \frac{ {x}^{2}+6x }{ {x}^{2}+\left( 4x \right)+4 }$

Write $8x$ as a sum

$\frac{ {x}^{2}+6x+2x+12 }{ {x}^{2}+12x+36 } \times \frac{ {x}^{2}+6x }{ {x}^{2}+\left( 4x \right)+4 }$

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

$\frac{ {x}^{2}+6x+2x+12 }{ {\left( x+6 \right)}^{2} } \times \frac{ {x}^{2}+6x }{ {x}^{2}+\left( 4x \right)+4 }$

Factor out $x$ from the expression

$\frac{ {x}^{2}+6x+2x+12 }{ {\left( x+6 \right)}^{2} } \times \frac{ x \times \left( x+6 \right) }{ {x}^{2}+\left( 4x \right)+4 }$

When there is a $+$ in front of an expression in parentheses, the expression remains the same

$\frac{ {x}^{2}+6x+2x+12 }{ {\left( x+6 \right)}^{2} } \times \frac{ x \times \left( x+6 \right) }{ {x}^{2}+4x+4 }$

Factor out $x$ from the expression

$\frac{ x \times \left( x+6 \right)+2x+12 }{ {\left( x+6 \right)}^{2} } \times \frac{ x \times \left( x+6 \right) }{ {x}^{2}+4x+4 }$

Factor out $2$ from the expression

$\frac{ x \times \left( x+6 \right)+2\left( x+6 \right) }{ {\left( x+6 \right)}^{2} } \times \frac{ x \times \left( x+6 \right) }{ {x}^{2}+4x+4 }$

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

$\frac{ x \times \left( x+6 \right)+2\left( x+6 \right) }{ {\left( x+6 \right)}^{2} } \times \frac{ x \times \left( x+6 \right) }{ {\left( x+2 \right)}^{2} }$

Factor out $x+6$ from the expression

$\frac{ \left( x+6 \right) \times \left( x+2 \right) }{ {\left( x+6 \right)}^{2} } \times \frac{ x \times \left( x+6 \right) }{ {\left( x+2 \right)}^{2} }$

Cancel out the common factor $x+6$

$\frac{ x+2 }{ x+6 } \times \frac{ x \times \left( x+6 \right) }{ {\left( x+2 \right)}^{2} }$

Cancel out the common factor $x+2$

$\frac{ 1 }{ x+6 } \times \frac{ x \times \left( x+6 \right) }{ x+2 }$

Cancel out the common factor $x+6$

$1 \times \frac{ x }{ x+2 }$

Any expression multiplied by $1$ remains the same

$\frac{ x }{ x+2 }$