To simplify this expression, we first factor the two fractions individually.
start fraction x squared plus eight x plus fifteen over x minus four end fraction can be factored as:
start fraction (x+5)(x+3) over (x-4) end fraction
start fraction x squared minus sixteen over two x plus six end fraction can be factored as:
start fraction (x+4)(x-4) over 2(x+3) end fraction
Now we can multiply the two fractions together:
start fraction (x+5)(x+3) over (x-4) end fraction times start fraction (x+4)(x-4) over 2(x+3) end fraction
To simplify this product, we can cancel out common factors in the numerator and denominator:
start fraction (x+5)cancel(x+3) over (x-4) end fraction times start fraction (x+4)cancel(x-4) over 2cancel(x+3) end fraction
This leaves us with:
start fraction (x+5)(x+4) over 2(x-4) end fraction
Therefore,
start fraction x squared plus eight x plus fifteen over x minus four end fraction times start fraction x squared minus sixteen over two x plus six end fraction = start fraction (x+5)(x+4) over 2(x-4) end fraction.