Differentiate $(x^{2}-5x+8)(x^{3}+7x+9)$ using the product rule.

Let $y = (x^{2}-5x+8)(x^{3}+7x+9)$.
Let $u = x^{2}-5x+8$ and $v = x^{3}+7x+9$.
Then $y = uv$.
By the product rule,$\frac{dy}{dx} = \frac{du}{dx}v + u\frac{dv}{dx}$.
$\frac{du}{dx} = \frac{d}{dx}(x^{2}-5x+8) = 2x-5$.
$\frac{dv}{dx} = \frac{d}{dx}(x^{3}+7x+9) = 3x^{2}+7$.
Substituting these into the formula:
$\frac{dy}{dx} = (2x-5)(x^{3}+7x+9) + (x^{2}-5x+8)(3x^{2}+7)$.
Expanding the terms:
$(2x-5)(x^{3}+7x+9) = 2x^{4} + 14x^{2} + 18x - 5x^{3} - 35x - 45 = 2x^{4} - 5x^{3} + 14x^{2} - 17x - 45$.
$(x^{2}-5x+8)(3x^{2}+7) = 3x^{4} + 7x^{2} - 15x^{3} - 35x + 24x^{2} + 56 = 3x^{4} - 15x^{3} + 31x^{2} - 35x + 56$.
Adding the two expressions:
$\frac{dy}{dx} = (2x^{4} - 5x^{3} + 14x^{2} - 17x - 45) + (3x^{4} - 15x^{3} + 31x^{2} - 35x + 56)$.
$\frac{dy}{dx} = 5x^{4} - 20x^{3} + 45x^{2} - 52x + 11$.