How do you factor \[{{\left( {{x}^{2}}+8 \right)}^{2}}-36{{x}^{2}}\]?

Hint: Use the algebraic identity ${{m}^{2}}-{{n}^{2}}=\left( m+n \right)\left( m-n \right)$ to simplify the given bi-quadratic expression and write them as the product of two quadratic expressions. Now, use the middle term split method to factorize the two quadratic expressions obtained and write the given expression as the product of its four factors given as \[\left( x-a \right)\left( x-b \right)\left( x-c \right)\left( x-d \right)\], where ‘a’, ‘b’, ‘c’ and ‘b’ are called zeroes of the polynomial.

Complete step by step solution:Here, we have been provided with the quadratic equation: \[{{\left( {{x}^{2}}+8 \right)}^{2}}-36{{x}^{2}}\] and we are asked to factor it. That means we have to write the expression as a product of its four factors.
As we can see that the given expression is a bi-quadratic expression so we will get a total of four factors. The given expression can be written as:
\[\Rightarrow {{\left( {{x}^{2}}+8 \right)}^{2}}-36{{x}^{2}}={{\left( {{x}^{2}}+8 \right)}^{2}}-{{\left( 6x \right)}^{2}}\]
Now, using the algebraic identity ${{m}^{2}}-{{n}^{2}}=\left( m+n \right)\left( m-n \right)$, we can simplify the expression as:
\[\Rightarrow {{\left( {{x}^{2}}+8 \right)}^{2}}-36{{x}^{2}}=\left( {{x}^{2}}+8+6x \right)\left( {{x}^{2}}+8-6x \right)\]
Rearranging the terms of the above two quadratic expressions, we have,
\[\Rightarrow {{\left( {{x}^{2}}+8 \right)}^{2}}-36{{x}^{2}}=\left( {{x}^{2}}+6x+8 \right)\left( {{x}^{2}}-6x+8 \right)\]
Now, using the middle term split method to factorize the two quadratic expressions, we get,
\[\begin{align}
  & \Rightarrow {{\left( {{x}^{2}}+8 \right)}^{2}}-36{{x}^{2}}=\left( {{x}^{2}}+2x+4x+8 \right)\left( {{x}^{2}}-2x-4x+8 \right) \\
 & \Rightarrow {{\left( {{x}^{2}}+8 \right)}^{2}}-36{{x}^{2}}=\left[ x\left( x+2 \right)+4\left( x+2 \right) \right]\left[ x\left( x-2 \right)-4\left( x-2 \right) \right] \\
 & \Rightarrow {{\left( {{x}^{2}}+8 \right)}^{2}}-36{{x}^{2}}=\left[ \left( x+2 \right)\left( x+4 \right) \right]\left[ \left( x-2 \right)\left( x-4 \right) \right] \\
 & \Rightarrow {{\left( {{x}^{2}}+8 \right)}^{2}}-36{{x}^{2}}=\left( x-2 \right)\left( x-4 \right)\left( x+2 \right)\left( x+4 \right) \\
\end{align}\]
Hence, the above relation is the factored form of the given expression.

Note: One may note that here we have considered the given expression as the bi-quadratic expression. This is because if we will simplify it using the algebraic identity ${{\left( m+n \right)}^{2}}={{m}^{2}}+{{n}^{2}}+2mn$ then we will see that the highest exponent of the variable x is 4 and such an expression is called the bi-quadratic expression. This is the reason we are getting four factors of the expression, two for each quadratic term. Remember the middle term split method which is used to factorize the quadratic expression or you may apply completing the square method.