How do you factor \[{x^2} + 3x - 10\] by using any mathematical method and elaborate the method with an example(take any general algebraic equation)?

Hint: By using the factorization method to obtain the factors of the given equation \[{x^2} + yx + z\] . Here we can use the mid term splitting method to factorise this equation in which we are going to split the middle term and can have required two factors.
For equation \[{x^2} + yx + z\]
Here we have to split “y” in such a way that the split number say “a” and “b” have the relationship \[a \times b = z(1)\] and \[a + b = y\] for the above equation.

Complete step-by-step answer:
The given equation is \[{x^2} + 3x - 10\]
Here, for the given equation split “3” into two number following the condition of midterm splitting rule-
 \[
   \Rightarrow {x^2} + 3x - 10 \\
   \Rightarrow {x^2} + (5 - 2)x - 10 \\
   \Rightarrow {x^2} + 5x - 2x - 10 \\
   \Rightarrow x(x + 5) - 2(x + 5) \\
   \Rightarrow (x + 5)(x - 2) \;
 \]
So the factors for the above equation is \[(x + 5),(x - 2)\]
So, the correct answer is “ \[(x + 5),(x - 2)\] ”.

Note: Mid term split method is easy to find factor, but another method that is taking the highest common factor directly and obtaining the factors. This method works sometimes in some questions only mostly for 3 variable questions.
For equation \[{x^2} + yx + z\]
Here we have to split “y” in such a way that the split number say “a” and “b” have the relationship \[a \times b = z(1)\] and \[a + b = y\] for the above equation.