Solve for: (x^2-6x-7)/(2x^2-98) \times (x^2+14x+49)/(4x^2+32x+28)

Expression: $\frac{ {x}^{2}-6x-7 }{ 2{x}^{2}-98 } \times \frac{ {x}^{2}+14x+49 }{ 4{x}^{2}+32x+28 }$

Write $-6x$ as a difference

$\frac{ {x}^{2}+x-7x-7 }{ 2{x}^{2}-98 } \times \frac{ {x}^{2}+14x+49 }{ 4{x}^{2}+32x+28 }$

Factor out $2$ from the expression

$\frac{ {x}^{2}+x-7x-7 }{ 2\left( {x}^{2}-49 \right) } \times \frac{ {x}^{2}+14x+49 }{ 4{x}^{2}+32x+28 }$

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

$\frac{ {x}^{2}+x-7x-7 }{ 2\left( {x}^{2}-49 \right) } \times \frac{ {\left( x+7 \right)}^{2} }{ 4{x}^{2}+32x+28 }$

Factor out $4$ from the expression

$\frac{ {x}^{2}+x-7x-7 }{ 2\left( {x}^{2}-49 \right) } \times \frac{ {\left( x+7 \right)}^{2} }{ 4\left( {x}^{2}+8x+7 \right) }$

Factor out $x$ from the expression

$\frac{ x \times \left( x+1 \right)-7x-7 }{ 2\left( {x}^{2}-49 \right) } \times \frac{ {\left( x+7 \right)}^{2} }{ 4\left( {x}^{2}+8x+7 \right) }$

Factor out $-7$ from the expression

$\frac{ x \times \left( x+1 \right)-7\left( x+1 \right) }{ 2\left( {x}^{2}-49 \right) } \times \frac{ {\left( x+7 \right)}^{2} }{ 4\left( {x}^{2}+8x+7 \right) }$

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

$\frac{ x \times \left( x+1 \right)-7\left( x+1 \right) }{ 2\left( x-7 \right) \times \left( x+7 \right) } \times \frac{ {\left( x+7 \right)}^{2} }{ 4\left( {x}^{2}+8x+7 \right) }$

Write $8x$ as a sum

$\frac{ x \times \left( x+1 \right)-7\left( x+1 \right) }{ 2\left( x-7 \right) \times \left( x+7 \right) } \times \frac{ {\left( x+7 \right)}^{2} }{ 4\left( {x}^{2}+7x+x+7 \right) }$

Factor out $x+1$ from the expression

$\frac{ \left( x+1 \right) \times \left( x-7 \right) }{ 2\left( x-7 \right) \times \left( x+7 \right) } \times \frac{ {\left( x+7 \right)}^{2} }{ 4\left( {x}^{2}+7x+x+7 \right) }$

Factor out $x$ from the expression

$\frac{ \left( x+1 \right) \times \left( x-7 \right) }{ 2\left( x-7 \right) \times \left( x+7 \right) } \times \frac{ {\left( x+7 \right)}^{2} }{ 4\left( x \times \left( x+7 \right)+x+7 \right) }$

Factor out $x+7$ from the expression

$\frac{ \left( x+1 \right) \times \left( x-7 \right) }{ 2\left( x-7 \right) \times \left( x+7 \right) } \times \frac{ {\left( x+7 \right)}^{2} }{ 4\left( x+7 \right) \times \left( x+1 \right) }$

Cancel out the common factor $x-7$

$\frac{ \left( x+1 \right) \times 1 }{ 2\left( x+7 \right) } \times \frac{ {\left( x+7 \right)}^{2} }{ 4\left( x+7 \right) \times \left( x+1 \right) }$

Cancel out the common factor $x+1$

$\frac{ 1 }{ 2\left( x+7 \right) } \times \frac{ {\left( x+7 \right)}^{2} }{ 4\left( x+7 \right) }$

Cancel out the common factor $x+7$

$\frac{ 1 }{ 2\left( x+7 \right) } \times \frac{ x+7 }{ 4 }$

Cancel out the common factor $x+7$

$\frac{ 1 }{ 2 } \times \frac{ 1 }{ 4 }$

Multiply the fractions

$\begin{align*}&\frac{ 1 }{ 8 } \\&\begin{array} { l }0.125,& {2}^{-3}\end{array}\end{align*}$