By Rational Root Theorem, all rational roots of a polynomial are in the form $\frac{p}{q}$, where $p$ divides the constant term $-64$ and $q$ divides the leading coefficient $1$. List all candidates $\frac{p}{q}$.
$$±64,±32,±16,±8,±4,±2,±1$$
Find one such root by trying out all the integer values, starting from the smallest by absolute value. If no integer roots are found, try out fractions.
$$a=4$$
By Factor theorem, $a-k$ is a factor of the polynomial for each root $k$. Divide $a^{3}-64$ by $a-4$ to get $a^{2}+4a+16$. Solve the equation where the result equals to $0$.
$$a^{2}+4a+16=0$$
All equations of the form $ax^{2}+bx+c=0$ can be solved using the quadratic formula: $\frac{-b±\sqrt{b^{2}-4ac}}{2a}$. Substitute $1$ for $a$, $4$ for $b$, and $16$ for $c$ in the quadratic formula.
