How do you factor ${x^3} + 8{x^2} - 20x?$

Hint: In this question we are going to find the factors of the given expression.
First take the common term outside from the given expression.
Next we are going to factor the expression by splitting the middle term.
Multiply the coefficient of the first term by a constant in the given expression, we get a number and then find two factors for that number whose sum equals the coefficient of the middle term.
Now rewrite the polynomial by splitting the middle term using the two factors that found before and then add the first two terms and last two terms, taking common factors outside from the first and last two terms. Add the four terms of the above step and we get the desired factorization.

Complete step-by-step solution:
In this question, we are going to factor the given expression and then find the values of $x$.
First write the given expression and mark it as $\left( 1 \right)$.
$ \Rightarrow {x^3} + 8{x^2} - 20x...\left( 1 \right)$
$x$ is common in all the terms, so take $x$ out from all the terms we get,
$ \Rightarrow x\left( {{x^2} + 8x - 20} \right)$
The above expression is of the quadratic form $a{x^2} + bx + c = 0$
Here the first term is ${x^2}$and its coefficient is $1$
The middle term is $8x$and its coefficient is $8$
The last term is $ - 20$and it is a constant.
First we are going to multiply the coefficient of first term by the last term.
That is, $1 \times - 20 = - 20$
Next we are going to find factors of $ - 20$ whose sum is equal to $8$
$10 - 2 = 8$
By splitting the middle term using the factors $10$ and $ - 2$ in the given expression
$x({x^2} + 10x - 2x - 20)$
Taking common factors outside from the two pairs
$ \Rightarrow x[x\left( {x + 10} \right) - 2\left( {x + 10} \right)]$
On rewriting we get,
$ \Rightarrow x[\left( {x - 2} \right)\left( {x + 10} \right)]$
Let us equating the term and we get
$ \Rightarrow x\left( {x - 2} \right) = 0,x\left( {x + 10} \right) = 0$
Let us multiply the term and we get
$ \Rightarrow {x^2} - 2x = 0,{x^2} + 10x = 0$
On rewriting we get
$ \Rightarrow {x^2} = 2x,{x^2} = - 10x$
Then we get,
$ \Rightarrow x = 2,x = - 10$
We get,
$ \Rightarrow x = 2, - 10$

The required factors of the expression ${x^3} + 8{x^2} - 20x$ are $x[\left( {x - 2} \right)\left( {x + 10} \right)]$.

Note: The following are some of the factoring methods to solve the expression: factoring out the GCF, the sum product pattern, the grouping method, the perfect square trinomial pattern, the difference of square pattern.
We can check our factoring by multiplying them all out to see if we get the original expression. If we do, our factoring is correct, otherwise we had to try again.