$\frac{ 4\left( {x}^{2}-1 \right) }{ {x}^{7} }\div\frac{ x-1 }{ {x}^{4} }$
To divide by a fraction, multiply by the reciprocal of that fraction$\frac{ 4\left( {x}^{2}-1 \right) }{ {x}^{7} } \times \frac{ {x}^{4} }{ x-1 }$
Use ${a}^{2}-{b}^{2}=\left( a-b \right)\left( a+b \right)$ to factor the expression$\frac{ 4\left( x-1 \right) \times \left( x+1 \right) }{ {x}^{7} } \times \frac{ {x}^{4} }{ x-1 }$
Cancel out the common factor ${x}^{4}$$\frac{ 4\left( x-1 \right) \times \left( x+1 \right) }{ {x}^{3} } \times \frac{ 1 }{ x-1 }$
Cancel out the common factor $x-1$$\frac{ 4\left( x+1 \right) }{ {x}^{3} }$
Distribute $4$ through the parentheses$\frac{ 4x+4 }{ {x}^{3} }$