Certainly! Let's factorize each of the given expressions step-by-step:
### (a) [tex]\( 5x + 10x(b + c) \)[/tex]
1. Identify common factors:
Both terms in the expression contain the factor [tex]\( 5x \)[/tex].
2. Factor out the common factor [tex]\( 5x \)[/tex]:
[tex]\[
5x + 10x(b + c) = 5x \left(1 + 2(b + c)\right)
\][/tex]
3. Simplify inside the parentheses:
[tex]\[
5x \left(1 + 2b + 2c\right)
\][/tex]
Final factorized form:
[tex]\[
5x(2b + 2c + 1)
\][/tex]
### (b) [tex]\( 3xy - 6x(y - z) \)[/tex]
1. Distribute the second term:
[tex]\[
3xy - 6x(y - z) = 3xy - 6xy + 6xz
\][/tex]
2. Combine like terms:
[tex]\[
3xy - 6xy + 6xz = -3xy + 6xz
\][/tex]
3. Identify common factors:
Both terms contain the factor [tex]\( -3x \)[/tex].
4. Factor out the common factor [tex]\( -3x \)[/tex]:
[tex]\[
-3xy + 6xz = -3x(y - 2z)
\][/tex]
Final factorized form:
[tex]\[
-3x(y - 2z)
\][/tex]
### (c) [tex]\( 2x(7 + y) - 14x(y + 2) \)[/tex]
1. Distribute the second term:
[tex]\[
2x(7 + y) - 14x(y + 2) = 2x \cdot 7 + 2x \cdot y - 14x \cdot y - 14x \cdot 2
\][/tex]
[tex]\[
= 14x + 2xy - 14xy - 28x
\][/tex]
2. Combine like terms:
[tex]\[
14x + 2xy - 14xy - 28x = 14x - 28x + 2xy - 14xy = -14x - 12xy
\][/tex]
3. Identify common factors:
Both terms contain the factor [tex]\( -2x \)[/tex].
4. Factor out the common factor [tex]\( -2x \)[/tex]:
[tex]\[
-14x - 12xy = -2x(7 + 6y)
\][/tex]
Final factorized form:
[tex]\[
-2x(6y + 7)
\][/tex]
### (d) [tex]\( -3a(2 + b) + 18a(b - 1) \)[/tex]
1. Distribute the second term:
[tex]\[
-3a(2 + b) + 18a(b - 1) = -3a \cdot 2 - 3a \cdot b + 18a \cdot b - 18a \cdot 1
\][/tex]
[tex]\[
= -6a - 3ab + 18ab - 18a
\][/tex]
2. Combine like terms:
[tex]\[
-6a - 18a - 3ab + 18ab = -24a + 15ab
\][/tex]
3. Identify common factors:
Both terms contain the factor [tex]\( 3a \)[/tex].
4. Factor out the common factor [tex]\( 3a \)[/tex]:
[tex]\[
-24a + 15ab = 3a(-8 + 5b)
\][/tex]
Final factorized form:
[tex]\[
3a(5b - 8)
\][/tex]
### (e) [tex]\( -4y(x - 2) - 12y(3 - x) \)[/tex]
1. Notice that [tex]\( (3 - x) = -(x - 3) \)[/tex]:
[tex]\[
-4y(x - 2) - 12y(3 - x) = -4y(x - 2) + 12y(x - 3)
\][/tex]
2. Distribute the second term:
[tex]\[
-4y(x - 2) + 12y(x - 3) = -4yx + 8y + 12yx - 36y
\][/tex]
3. Combine like terms:
[tex]\[
-4yx + 12yx + 8y - 36y = 8yx - 28y
\][/tex]
4. Identify common factors:
Both terms contain the factor [tex]\( 4y \)[/tex].
5. Factor out the common factor [tex]\( 4y \)[/tex]:
[tex]\[
8yx - 28y = 4y(2x - 7)
\][/tex]
Final factorized form:
[tex]\[
4y(2x - 7)
\][/tex]
Thus, the fully factorized forms are:
(a) [tex]\(5x(2b + 2c + 1)\)[/tex]
(b) [tex]\(-3x(y - 2z)\)[/tex]
(c) [tex]\(-2x(6y + 7)\)[/tex]
(d) [tex]\(3a(5b - 8)\)[/tex]
(e) [tex]\(4y(2x - 7)\)[/tex]
