How do you long divide $\dfrac{{{a^4} + 4{b^4}}}{{{a^2} - 2ab + 2{b^2}}}$

Hint: Given the polynomials at the numerator and denominator, and we have to divide the numerator by the denominator. To divide the polynomials, the long division method is used in which the terms of the numerator are written as dividend and the terms of the denominator are written as the divisor. Then, write zero for missing terms. Then, write the first term of the quotient so that it will multiply by the divisor and matches with the first term of the dividend. Then, write the result of multiplication under the dividend and subtract the terms. Then, bring down the next term from the dividend and again write the quotient such that it matches with the leading term of the remainder.

Complete step-by-step answer:
We are given the polynomial. Apply the long division method to divide the numerator by the denominator. Add the coefficient zero for the missing terms.
\[
  {\text{ }}\underline {{\text{ }}} \\
  {a^2} - 2ab + 2{b^2}){a^4} + 0{a^3}b + 0{a^2}{b^2} + 0a{b^3} + 4{b^4} \\
 \]
Now, divide the first term of the dividend by the first term of divisor to choose the quotient of the division.
 $ \Rightarrow \dfrac{{{a^4}}}{{{a^2}}} = {a^2}$
So, the first term of the quotient is ${a^2}$ and multiply the divisor by ${a^2}$and write under the dividend and subtract the terms.
\[
  {\text{ }}\underline {{\text{ }}{a^2}{\text{ }}} \\
  {a^2} - 2ab + 2{b^2}){a^4} + 0{a^3}b + 0{a^2}{b^2} + 0a{b^3} + 4{b^4} \\
  {\text{ }}\underline {( - ){a^4} - 2{a^3}b + 2{a^2}{b^2}{\text{ }}} \\
  {\text{ }}2{a^3}b - 2{a^2}{b^2} \\
 \]
Now, again we will divide the leading term of the remainder by the first term of divisor to find the next term of the quotient.
$ \Rightarrow \dfrac{{2{a^3}b}}{{{a^2}}} = 2ab$
So, the next term of the quotient is $2ab$ and multiply the divisor by $2ab$ and write under the dividend and subtract the terms.
\[
  {\text{ }}\underline {{\text{ }}{a^2} + 2ab{\text{ }}} \\
  {a^2} - 2ab + 2{b^2}){a^4} + 0{a^3}b + 0{a^2}{b^2} + 0a{b^3} + 4{b^4} \\
  {\text{ }}\underline {( - ){a^4} - 2{a^3}b + 2{a^2}{b^2}{\text{ }}} \\
  {\text{ }}2{a^3}b - 2{a^2}{b^2} + 0a{b^3} \\
  {\text{ }}\underline {( - )2{a^3}b - 4{a^2}{b^2} + 4a{b^3}} \\
  {\text{ }}2{a^2}{b^2} - 4a{b^3} \\
 \]
Now we will bring down the next term of the dividend. Then we will divide the leading term of the remainder by the first term of divisor to find the next term of the quotient.
$ \Rightarrow \dfrac{{2{a^2}{b^2}}}{{{a^2}}} = 2{b^2}$
So, the next term of the quotient is $2{b^2}$ and multiply the divisor by $2{b^2}$and write under the dividend and subtract the terms.
\[
  {\text{ }}\underline {{\text{ }}{a^2} + 2ab + 2{b^2}} \\
  {a^2} - 2ab + 2{b^2}){a^4} + 0{a^3}b + 0{a^2}{b^2} + 0a{b^3} + 4{b^4} \\
  {\text{ }}\underline {( - ){a^4} - 2{a^3}b + 2{a^2}{b^2}{\text{ }}} \\
  {\text{ }}2{a^3}b - 2{a^2}{b^2} + 0a{b^3} \\
  {\text{ }}\underline {( - )2{a^3}b - 4{a^2}{b^2} + 4a{b^3}{\text{ }}} \\
  {\text{ }}2{a^2}{b^2} - 4a{b^3} + 4{b^4} \\
  {\text{ }}\underline {( - )2{a^2}{b^2} - 4a{b^3} + 4{b^4}} \\
  {\text{ }}0 \\
 \]
Thus, the remainder of the division is \[0\]and the quotient of the division is \[{a^2} + 2ab + 2{b^2}\]

Final answer: Hence the result of the long division is \[{a^2} + 2ab + 2{b^2}\]

Note:
In such types of questions students mainly make mistakes while choosing the quotient of the division which when multiplied with the terms of the divisor, we will get the term that matches with the first term of the dividend.