The problem asks us to expand the expression $-t(-4t^3 + 8t^2)$. We need to multiply the monomial $-t$ by the polynomial $(-4t^3 + 8t^2)$ and express the result as a polynomial in standard form.

1. Problem Description

The problem asks us to expand the expression $-t(-4t^3 + 8t^2)$. We need to multiply the monomial $-t$ by the polynomial $(-4t^3 + 8t^2)$ and express the result as a polynomial in standard form.

2. Solution Steps

We will use the distributive property to multiply $-t$ by each term inside the parentheses.

$-t(-4t^3 + 8t^2) = (-t)(-4t^3) + (-t)(8t^2)$

Recall the rule for multiplying exponents: $a^m \cdot a^n = a^{m+n}$.

$(-t)(-4t^3) = (-1 \cdot t^1)(-4 \cdot t^3) = (-1)(-4)(t^1 \cdot t^3) = 4t^{1+3} = 4t^4$.

$(-t)(8t^2) = (-1 \cdot t^1)(8 \cdot t^2) = (-1)(8)(t^1 \cdot t^2) = -8t^{1+2} = -8t^3$.

Therefore, $-t(-4t^3 + 8t^2) = 4t^4 - 8t^3$.

3. Final Answer

$4t^4 - 8t^3$