$=\,\,$ $64 \times x^3+1$
$=\,\,$ $4 \times 16 \times x^3+1$
$=\,\,$ $4 \times 4 \times 4 \times x^3+1$
$=\,\,$ $4^3 \times x^3+1$
$=\,\,$ $(4 \times x)^3+1$
$=\,\,$ $(4x)^3+1$
$=\,\,$ $(4x)^3+1 \times 1 \times 1$
$=\,\,$ $(4x)^3+1^3$
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