The problem requires simplifying the expression $4(3x^2 + 5x) - x(x^2 - 7x + 8)$.

1. Problem Description

The problem requires simplifying the expression $4(3x^2 + 5x) - x(x^2 - 7x + 8)$.

2. Solution Steps

First, distribute the $4$ across the terms inside the first parentheses:

$4(3x^2 + 5x) = 4(3x^2) + 4(5x) = 12x^2 + 20x$

Next, distribute the $-x$ across the terms inside the second parentheses:

$-x(x^2 - 7x + 8) = -x(x^2) - x(-7x) - x(8) = -x^3 + 7x^2 - 8x$

Now, combine the results:

$12x^2 + 20x - x^3 + 7x^2 - 8x$

Finally, group like terms together:

$-x^3 + (12x^2 + 7x^2) + (20x - 8x)$

Combine the like terms:

$-x^3 + 19x^2 + 12x$

3. Final Answer

$-x^3 + 19x^2 + 12x$