How do you factor $2x(2x + 5) + ({x^2})2$?

Hint: In this question, we have to solve the given expression. When we solve the given expression, we will get the equation in quadratic form. The general form of the quadratic equation is $a{x^2} + bx + c = 0$. Where ‘a’ is the coefficient of${x^2}$, ‘b’ is the coefficient of x and ‘c’ is the constant term.
To solve this equation, we will apply the sum-product pattern. During the simplification, we will take out common factors from the two pairs. Then we will rewrite it in factored form.
Therefore, we should follow the below steps:
Apply sum-product patterns.
Make two pairs.
Common factor from two pairs.
Rewrite in factored form.

Complete step by step solution:
In this question, the given expression is:
$ \Rightarrow 2x(2x + 5) + ({x^2})2$
Let us simplify the above expression,
$ \Rightarrow 4{x^2} + 10x + 2{x^2}$
Let us add the coefficient of ${x^2}$that is 4 and 2. The answer is 6.
$ \Rightarrow 6{x^2} + 10x$
Here, we get the quadratic equation. In the above quadratic equation, the constant term is 0. Therefore, we can solve it directly by taking out the common factor from the equation.
Let us take out the common factor.
$ \Rightarrow 2x\left( {3x + 5} \right)$

Hence, the solution of the given quadratic equation is $2x\left( {3x + 5} \right)$.

Note:
One important thing is, we can always check our work by multiplying out factors back together, and check that we have got back the original answer.
To check our factorization, multiplication goes like this:
$ \Rightarrow 2x\left( {3x + 5} \right)$
Let us apply multiplication to remove brackets.
$ \Rightarrow 6{x^2} + 10x$
Hence, we get our quadratic equation back by applying multiplication.
Here is a list of methods to solve quadratic equations:
Factorization
Completing the square
Using graph
Quadratic formula