Factor the polynomial: 64x^3 + 343
Set it equal to zero and find the roots, that will give you a factor!
$\displaystyle 64x^3+343 = 0 $
$\displaystyle x = (\frac{-343}{64})^{\frac{1}{3}} = -\frac{7}{4} $
Hence $\displaystyle x +\frac{7}{4} $ is a factor.
Use long division to find $\displaystyle f(x)$ such that$\displaystyle (x+\frac{7}{4})f(x) = 64x^3+343$
You should get $\displaystyle (x+\frac{7}{4})(64x^2+112x+196) = 64x^3+343$
The quadratic in this equation is irreducible.
You have (4x)^3 + 7^3. Now use the sum of cubes formula for factorising: Sum of Cubes
Note that the quadratic factor in this formula cannot be further factorised.