How do you factor ${x^4} + 6{x^2} - 7$ completely?

Hint: In order to factor the expression, we first make it into the standard quadratic equation form by turning ${x^4}$ into ${\left( {{x^2}} \right)^2}$ thus our equation becomes: ${\left( {{x^2}} \right)^2} + 6{x^2} - 7$ , then we use grouping method to factorize the remaining equation. We split the middle term into two terms according to the sum- product rule, which states that the middle term should be split in such a way that the two terms are product of the first and the last numbers, and the sum of the terms gives us the original middle term. We solve it further and get our required factors and answer.

Complete step-by-step solution:
Given expression is ${x^4} + 6{x^2} - 7$
Let us turn the given equation into standard quadratic equation form so that we can solve it easily:
Thus, ${x^4} + 6{x^2} - 7 = {\left( {{x^2}} \right)^2} + 6{x^2} - 7$
Therefore, our new equation becomes: ${\left( {{x^2}} \right)^2} + 6{x^2} - 7$
Now, in order to solve the equation further we need to factorize it.
Here the middle term is $ + 6{x^2}$ , we need to split the middle term into two terms in such a way that the product of those terms is equal to the product of the first term $\left( {{x^4}} \right)$ and third term $\left( { - 7} \right)$ and the sum of the two terms gives us the original middle term.
The product of the first term and third terms is $ - 7{x^4}$ , in order to split the middle term, we find the factors of the $ - 7{x^4}$ .
Factors of $ - 7{x^4}$ which when added together will give us the middle term $6{x^2}$ are $7,1$ .
Thus, we split the middle term: $6{x^2} = 7{x^2} - {x^2}$
Placing the above mentioned middle term in our quadratic equation, we get:
$ \Rightarrow {\left( {{x^2}} \right)^2} + 7{x^2} - {x^2} - 7$
Now, we need to make two groups and find common factors. We first arrange our quadratic equation so that we could do the groupings easily:
$ \Rightarrow {\left( {{x^2}} \right)^2} - {x^2} + 7{x^2} - 7$
Now, we find the factors:
$ \Rightarrow {x^2}\left( {{x^2} - 1} \right) + 7\left( {{x^2} - 1} \right)$
Here $\left( {{x^2} - 1} \right)$ is a common factor. We further solve it:
$\left( {{x^2} + 7} \right)\left( {{x^2} - 1} \right)$
We know that: $\left( {{a^2} - {b^2}} \right) = \left( {a + b} \right)\left( {a - b} \right)$ .
Therefore, we can expand $\left( {{x^2} - 1} \right)$ as $\left( {x + 1} \right)\left( {x - 1} \right)$ .

Thus, our factors are: $\left( {{x^2} + 7} \right)\left( {x + 1} \right)\left( {x - 1} \right)$.

Note: Factorization is simply the method of breaking down a given expression into their simplest factors. Grouping method used to solve the sum above is the most common method of solving any quadratic equation. One should also be attentive towards the signs placed in front of the numbers, as many students make a mistake in getting confused with the signs.