How do you factor $64{{x}^{3}}+343$?

Hint: We can observe that the given equation contains one cubic variable in addition. In algebra we have the formula ${{a}^{3}}+{{b}^{3}}=\left( a+b \right)\left( {{a}^{2}}-ab+{{b}^{2}} \right)$. But in the given equation we have additionally coefficient for the variable and a constant, so we need to factorize the coefficient of the given variable, constant at the same time and we will convert the given equation in form of ${{a}^{3}}+{{b}^{3}}$, by applying some exponential rules, then we will use the algebraic formula and write the given equation as the product of its factors.

Complete step-by-step solution:

Given equation $64{{x}^{3}}+343$.
Considering the coefficient of ${{x}^{3}}$, which is $64$. Factoring $64$.
Dividing $64$ with $2$, then we will get zero remainder and $32$ as quotient. So, we can write $64$ as
$64=2\times 32$.
Now we will factorize $32$. We know that $32$ is divisible by $2$. We will get zero reminder and $16$ as quotient, then we can write $32$ as
$32=2\times 16$
Now we will factorize $16$. We know that we can write $16$ as $16=4\times 4$. From all the above values, the value of $64$ can be
$\begin{align}
  & 64=2\times 32 \\
 & \Rightarrow 64=2\times 2\times 16 \\
 & \Rightarrow 64=2\times 2\times 4\times 4 \\
 & \Rightarrow 64=4\times 4\times 4 \\
\end{align}$
We have an exponential rule $a\times a\times a\times ....n\text{ times}={{a}^{n}}$, then we will have
$\therefore 64={{4}^{3}}$
Considering the constant which is $343$. Factoring $343$.
We know that we can write $343$ as $343=7\times 7\times 7$
We have an exponential rule $a\times a\times a\times ....n\text{ times}={{a}^{n}}$, then we will have
$\therefore 343={{7}^{3}}$.
Now the given equation is modified as
$64{{x}^{3}}+343={{4}^{3}}{{x}^{3}}+{{7}^{3}}$
We have the exponential rule ${{a}^{m}}{{b}^{m}}={{\left( ab \right)}^{m}}$, then we will get
$\Rightarrow 64{{x}^{3}}+343={{\left( 4x \right)}^{3}}+{{7}^{3}}$
Applying the formula ${{a}^{3}}+{{b}^{3}}=\left( a+b \right)\left( {{a}^{2}}-ab+{{b}^{2}} \right)$ in the above equation, then we will get
$\begin{align}
  & \Rightarrow 64{{x}^{3}}+343=\left( 4x+7 \right)\left( {{\left( 4x \right)}^{2}}-4x\left( 7 \right)+{{7}^{2}} \right) \\
 & \Rightarrow 64{{x}^{3}}+343=\left( 4x+7 \right)\left( 16{{x}^{2}}-28x+49 \right) \\
\end{align}$

Note: When we multiply the above obtained factors, we need to get the given equation as result. If you don’t get the given equation as a result, then our solution is not correct. If they are the same then our result is correct.