(a-4b)(a²+3ab-6b²)

To expand the expression \((a - 4b)(a^2 + 3ab - 6b^2)\), we can use the distributive property (also known as the FOIL method for binomials, although here we have a binomial and a trinomial).

We will distribute \(a\) and \(-4b\) to each term in the trinomial:

1. Distributing \(a\): \[ a \cdot a^2 = a^3 \] \[ a \cdot 3ab = 3a^2b \] \[ a \cdot (-6b^2) = -6ab^2 \]

2. Distributing \(-4b\): \[ -4b \cdot a^2 = -4a^2b \] \[ -4b \cdot 3ab = -12ab^2 \] \[ -4b \cdot (-6b^2) = 24b^3 \]

Now, we can combine all these results: \[ a^3 + 3a^2b - 6ab^2 - 4a^2b - 12ab^2 + 24b^3 \]

Next, we combine like terms:

• The \(a^2b\) terms: \(3a^2b - 4a^2b = -a^2b\)
• The \(ab^2\) terms: \(-6ab^2 - 12ab^2 = -18ab^2\)

Combining everything gives us: \[ a^3 - a^2b - 18ab^2 + 24b^3 \]

Thus, the expanded expression is: \[ \boxed{a^3 - a^2b - 18ab^2 + 24b^3} \]