Factorize \[{x^2} - 32x - 105\].

Hint: In the question we are asked to factorize the given quadratic expression. So, we will use the middle term splitting method to factorize it. In this method we split the middle term that is the term containing \[x\] in such a way that its sum is equal to the middle term and its product is equal to the product of the term containing \[{x^2}\] and the constant term.

Complete step by step answer:
Given expression:
\[{x^2} - 32x - 105\]
Here the middle term is \[ = - 32x\]
Constant term \[ = - 105\]
Product of term with \[{x^2}\] and constant term \[ = {x^2} \times \left( { - 105} \right)\]
\[ = - 105{x^2}\]
So, we will split the middle term that is \[ - 32x\] in such a way that its sum is equal to \[ - 32x\] and the product is equal to \[ - 105{x^2}\].
Now,
\[ \Rightarrow {x^2} - 32x - 105 = {x^2} - 35x + 3x - 105\]
Here, we can see that
\[ - 35x + 3x = - 32x\] and,
\[ - 35x \times 3x = - 105{x^2}\]
Now, we will take out the common terms.
\[ \Rightarrow {x^2} - 32x - 105 = x\left( {x - 35} \right) + 3\left( {x - 35} \right)\]
Now we will take \[\left( {x - 35} \right)\] common.
\[ \Rightarrow {x^2} - 32x - 105 = \left( {x - 35} \right)\left( {x + 3} \right)\]
Hence, \[\left( {x - 35} \right){\text{ and }}\left( {x + 3} \right)\] are the factors of \[{x^2} - 32x - 105\].

Note:
Alternative approach:
We can also solve this question by hit and trial method. In this method first we will find one of the roots of the given expression by hit and trial method.
Root of an equation is the value of a variable for which the given equation becomes equal to zero.
Given expression:
\[{x^2} - 32x - 105\]
By hit and trial, that is by putting different numerical values of \[x\] we will check whether the given expression becomes zero or not.
So, when we put \[x = - 3\] we see that the above expression becomes zero.
Let’s put \[x = - 3\], the expression becomes,
\[ = {\left( { - 3} \right)^2} - 32\left( { - 3} \right) - 105\]
\[ = 9 + 96 - 105\]
\[ = 0\]
So, it means that \[x = - 3\] is a root of the above expression. Hence, \[\left( {x + 3} \right)\] is one of the two factors of the given expression. Now, to find the other factor we will divide \[{x^2} - 32x - 105\] by \[\left( {x + 3} \right)\] according to the polynomial division method.
On division we get \[\left( {x - 35} \right)\] as the quotient and remainder equal to zero.
So, we can write,
\[{x^2} - 32x - 105 = \left( {x + 3} \right)\left( {x - 35} \right) + 0\]
\[ \Rightarrow {x^2} - 32x - 105 = \left( {x + 3} \right)\left( {x - 35} \right)\]