Certainly! Let's factor each of the given expressions step-by-step:
### 1. Factor [tex]\( 2bd - 8d + 5b - 20 \)[/tex]
- First, we group the terms to make factoring by grouping easier:
[tex]\( 2bd - 8d + 5b - 20 = 2d(b - 4) + 5(b - 4) \)[/tex]
- Notice that [tex]\( b - 4 \)[/tex] is a common factor:
[tex]\[
2d(b - 4) + 5(b - 4) = (b - 4)(2d + 5)
\][/tex]
- Therefore, the factored form is:
[tex]\[
2bd - 8d + 5b - 20 = (b - 4)(2d + 5)
\][/tex]
### 2. Factor [tex]\( 2xy + 2x + 4y + 8 \)[/tex]
- First, we group the terms to make factoring easier:
[tex]\( 2xy + 2x + 4y + 8 = 2x(y + 1) + 4(y + 2) \)[/tex]
- Notice that [tex]\( 2 \)[/tex] is a common factor:
[tex]\[
2(xy + x + 2y + 4)
\][/tex]
- Therefore, the factored form is:
[tex]\[
2(xy + x + 2y + 4)
\][/tex]
### 3. Factor [tex]\( mn + 2m + 2n + 4 \)[/tex]
- First, we group the terms to make factoring easier:
[tex]\( mn + 2m + 2n + 4 = m(n + 2) + 2(n + 2) \)[/tex]
- Notice that [tex]\( n + 2 \)[/tex] is a common factor:
[tex]\[
m(n + 2) + 2(n + 2) = (m + 2)(n + 2)
\][/tex]
- Therefore, the factored form is:
[tex]\[
mn + 2m + 2n + 4 = (m + 2)(n + 2)
\][/tex]
### 4. Factor [tex]\( 4mn - 4m + 3n - 12 \)[/tex]
- Observe the polynomial:
[tex]\[
4mn - 4m + 3n - 12
\][/tex]
- Group the terms and attempt to factor by grouping:
[tex]\[
4mn - 4m + 3n - 12
\][/tex]
- Factoring attempts show that this expression does not factor neatly into common binomial or monomial terms.
- Therefore, in its essentially factored form, it is:
[tex]\[
4mn - 4m + 3n - 12
\][/tex]
### 5. Factor [tex]\( ax - ay + bx - by \)[/tex]
- First, we group the terms to make factoring easier:
[tex]\( ax - ay + bx - by = a(x - y) + b(x - y) \)[/tex]
- Notice that [tex]\( x - y \)[/tex] is a common factor:
[tex]\[
a(x - y) + b(x - y) = (a + b)(x - y)
\][/tex]
- Therefore, the factored form is:
[tex]\[
ax - ay + bx - by = (a + b)(x - y)
\][/tex]
Thus, the fully factored expressions are:
1. [tex]\( (b - 4)(2d + 5) \)[/tex]
2. [tex]\( 2(xy + x + 2y + 4) \)[/tex]
3. [tex]\( (m + 2)(n + 2) \)[/tex]
4. [tex]\( 4mn - 4m + 3n - 12 \)[/tex]
5. [tex]\( (a + b)(x - y) \)[/tex]
