[tex]$\frac{(x+4)^{2}}{x-4} \div \frac{x^{2}-16}{4 x-16}=\frac{4(x+4)}{(x-4)}[/tex]
Solution:
Given expression is
[tex]$\frac{(x+4)^{2}}{x-4} \div \frac{x^{2}-16}{4 x-16}[/tex]
To solve this expression:
[tex]$\frac{(x+4)^{2}}{x-4} \div \frac{x^{2}-16}{4 x-16}=\frac{(x+4)^{2}}{x-4} \div \frac{x^{2}-4^2}{4( x-4)}[/tex]
Using algebraic identity: [tex]a^2-b^2=(a-b)(a+b)[/tex]
[tex]$=\frac{(x+4)^{2}}{x-4} \div \frac{(x-4)(x+4)}{4( x-4)}[/tex]
We can't solve it with division symbol. So change this into multiplication and solve it.
The second term is reversed when you change division into multiplication.
[tex]$=\frac{(x+4)^{2}}{x-4} \times \frac{4(x-4)}{( x-4)(x+4)}[/tex]
[tex]$=\frac{(x+4)(x+4)}{x-4} \times \frac{4(x-4)}{( x-4)(x+4)}[/tex]
[tex]$=\frac{4(x+4)(x+4)(x-4)}{(x-4)( x-4)(x+4)}[/tex]
Now, cancel the common terms in the numerator and denominator.
[tex]$=\frac{4(x+4)}{(x-4)}[/tex]
[tex]$\frac{(x+4)^{2}}{x-4} \div \frac{x^{2}-16}{4 x-16}=\frac{4(x+4)}{(x-4)}[/tex]
Hence the answer is [tex]\frac{4(x+4)}{(x-4)}[/tex].
