Question

1. $64c^3 + 1$

Explanation:

Step1: Recall the sum of cubes formula

The sum of cubes formula is \(a^3 + b^3=(a + b)(a^2 - ab + b^2)\). We can rewrite \(64c^3+1\) as \((4c)^3+1^3\).

Step2: Apply the sum of cubes formula

Here, \(a = 4c\) and \(b = 1\). Substituting into the formula:
\[

$$\begin{align*} (4c)^3+1^3&=(4c + 1)((4c)^2-(4c)\times1+1^2)\\ &=(4c + 1)(16c^2-4c + 1) \end{align*}$$

\]

Answer:

\((4c + 1)(16c^2-4c + 1)\)