ANSWER
[tex]\frac{x - 4}{2x - 2} [/tex]
EXPLANATION
The given expression is;
[tex] \frac{2 {x}^{2} + 5x + 3}{ {x}^{2} - 3x - 4} \div \frac{4 {x}^{2} + 2x - 6}{ {x}^{2} - 8x + 16} [/tex]
We factor to obtain;
[tex] \frac{(x + 1)(2x + 3)}{(x - 4)(x + 1)} \div \frac{2(x - 1)(2x + 3)}{(x - 4)(x - 4)} [/tex]
Multiply by the reciprocal of the second fraction
[tex]\frac{(x + 1)(2x + 3)}{(x - 4)(x + 1)} \times \frac{(x - 4)(x - 4)}{2(x - 1)(2x + 3)} [/tex]
Cancel out common factors to get,
[tex]\frac{1}{1} \times \frac{(x - 4)}{2(x - 1)} [/tex]
[tex]\frac{x - 4}{2x - 2} [/tex]
