Factories the expression
I.\[a{x^2} + bx\]
II. \[7{p^2} + 21{q^2}\]
III.\[2{x^3} + 2x{y^2} + 2x{z^2}\]
IV.\[a{m^2} + b{m^2} + b{n^2} + a{n^2}\]
V.\[\left( {lm + l} \right) + m + 1\]
VI.\[y\left( {y + z} \right) + 9\left( {y + z} \right)\]
VII.\[5{y^2} - 20y - 8z + 2yz\]
VIII.\[10ab + 4a + 5b + 2\]
IX.\[6xy - 4y + 6 - 9x\]

Hint: Factorization is the process of finding the factors of the expressions or the numbers. A number or quantity, when multiplied with another number, produces the given number or expression is known as the factor.
In this question, first, arrange the terms of the given expression with the term having the same variables and then find their common multiples and this will be the factors of the expressions. In other words, we can say that to separate the common terms of the given expression is known as factorization.

Complete step-by-step answer:
Start by arranging the term with the same variables or the multiple then find their common multiples,
I.\[a{x^2} + bx\]
In this expression, x is the common multiple; hence its factor is
\[a{x^2} + bx = x\left( {ax + b} \right)\]

II.\[7{p^2} + 21{q^2}\]
In this expression coefficient of variable’s are in multiple of 7; hence its factor is
\[7{p^2} + 21{q^2} = 7\left( {{p^2} + 3{q^2}} \right)\]

III.\[2{x^3} + 2x{y^2} + 2x{z^2}\]
In this expression, x and 2 is the common multiple; hence its factor is
\[2{x^3} + 2x{y^2} + 2x{z^2} = 2x\left( {{x^2} + {y^2} + {z^2}} \right)\]

IV.\[a{m^2} + b{m^2} + b{n^2} + a{n^2}\]
In this expression \[{m^2}\]and \[{n^2}\] is the common multiple; hence its factor is
\[a{m^2} + b{m^2} + b{n^2} + a{n^2} = {m^2}\left( {a + b} \right) + {n^2}\left( {a + b} \right)\]
In the factor \[\left( {a + b} \right)\]is two times hence we can write the factors as
\[
  a{m^2} + b{m^2} + b{n^2} + a{n^2} = {m^2}\left( {a + b} \right) + {n^2}\left( {a + b} \right) \\
   = \left( {a + b} \right)\left( {{m^2} + {n^2}} \right) \\
 \]

V.\[\left( {lm + l} \right) + m + 1\]
In this expression \[l\] is the common multiple; hence its factor is
\[
  \left( {lm + l} \right) + m + 1 = l\left( {m + 1} \right) + m + 1 \\
   = \left( {m + 1} \right)\left( {l + 1} \right) \\
 \]

VI.\[y\left( {y + z} \right) + 9\left( {y + z} \right)\]
In the expression \[\left( {y + z} \right)\]is two times hence we can write the factors as
\[y\left( {y + z} \right) + 9\left( {y + z} \right) = \left( {y + 9} \right)\left( {y + z} \right)\]

VII.\[5{y^2} - 20y - 8z + 2yz\]
In this expression, y and z is the common multiple; hence its factor is
\[
  5{y^2} - 20y - 8z + 2yz = 5{y^2} - 20y + 2yz - 8z \\
   = 5y\left( {y - 4} \right) + 2z\left( {y - 4} \right) \\
 \]
In the factor \[\left( {y - 4} \right)\]is two times hence we can write the factors as
\[
  5{y^2} - 20y - 8z + 2yz = 5y\left( {y - 4} \right) + 2z\left( {y - 4} \right) \\
   = \left( {5y + 2z} \right)\left( {y - 4} \right) \\
 \]

VIII.\[10ab + 4a + 5b + 2\]
In this expression, a is the common multiple; hence its factor is
\[10ab + 4a + 5b + 2 = 2a\left( {5b + 2} \right) + \left( {5b + 2} \right)\]
In the factor \[\left( {5b + 2} \right)\]is two times hence we can write the factors as
\[
  10ab + 4a + 5b + 2 = 2a\left( {5b + 2} \right) + \left( {5b + 2} \right) \\
   = \left( {2a + 1} \right)\left( {5b + 2} \right) \\
 \]

IX.\[6xy - 4y + 6 - 9x\]
Arrange the given expression in terms of the variables
\[6xy - 4y + 6 - 9x = 6xy - 9x - 4y + 6\]
In this expression 2, 3, and x are the common multiple; hence its factor is
\[
  6xy - 4y + 6 - 9x = 6xy - 9x - 4y + 6 \\
   = 3x\left( {2y - 3} \right) - 2\left( {2y - 3} \right) \\
 \]
In the factor \[\left( {2y - 3} \right)\]is two times hence we can write the factors as
\[
  6xy - 9x - 4y + 6 = 3x\left( {2y - 3} \right) - 2\left( {2y - 3} \right) \\
   = \left( {3x - 2} \right)\left( {2y - 3} \right) \\
 \]

Note: Students must note while finding the common multiple of the expressions, first arrange the terms of the expression in similar terms and then proceed with the factorization process.