Differentiate $\left(x^{2}-5 x+8\right)\left(x^{3}+7 x+9\right)$ by expanding the product to obtain a single polynomial.

Let $y = (x^{2}-5x+8)(x^{3}+7x+9)$.
First,expand the product:
$y = x^{2}(x^{3}+7x+9) - 5x(x^{3}+7x+9) + 8(x^{3}+7x+9)$
$y = x^{5} + 7x^{3} + 9x^{2} - 5x^{4} - 35x^{2} - 45x + 8x^{3} + 56x + 72$
Combine like terms:
$y = x^{5} - 5x^{4} + (7+8)x^{3} + (9-35)x^{2} + (-45+56)x + 72$
$y = x^{5} - 5x^{4} + 15x^{3} - 26x^{2} + 11x + 72$
Now,differentiate with respect to $x$:
$\frac{dy}{dx} = \frac{d}{dx}(x^{5} - 5x^{4} + 15x^{3} - 26x^{2} + 11x + 72)$
Using the power rule $\frac{d}{dx}(x^{n}) = nx^{n-1}$:
$\frac{dy}{dx} = 5x^{4} - 5(4x^{3}) + 15(3x^{2}) - 26(2x) + 11(1) + 0$
$\frac{dy}{dx} = 5x^{4} - 20x^{3} + 45x^{2} - 52x + 11$