Sure! Let's factorize each expression step-by-step:
### (a) [tex]\(2ax + 6ap + bx + 3bp\)[/tex]
1. Group the terms to facilitate factoring:
[tex]\[
2ax + 6ap + bx + 3bp = (2ax + 6ap) + (bx + 3bp)
\][/tex]
2. Factor out the common factors from each group:
[tex]\[
2a(x + 3p) + b(x + 3p)
\][/tex]
3. Notice that [tex]\((x + 3p)\)[/tex] is a common factor:
[tex]\[
(2a + b)(x + 3p)
\][/tex]
Thus, the factorized form is:
[tex]\[
2ax + 6ap + bx + 3bp = (2a + b)(x + 3p)
\][/tex]
### (b) [tex]\(xs - xk - ys + yk\)[/tex]
1. Group the terms to facilitate factoring:
[tex]\[
xs - xk - ys + yk = (xs - xk) - (ys - yk)
\][/tex]
2. Factor out the common factors from each group:
[tex]\[
x(s - k) - y(s - k)
\][/tex]
3. Notice that [tex]\((s - k)\)[/tex] is a common factor:
[tex]\[
(s - k)(x - y)
\][/tex]
Thus, the factorized form is:
[tex]\[
xs - xk - ys + yk = (s - k)(x - y)
\][/tex]
### (c) [tex]\(6ma - nb - 3na + 2mb\)[/tex]
1. Group the terms to facilitate factoring:
[tex]\[
6ma - 3na - nb + 2mb = (6ma - 3na) + (-nb + 2mb)
\][/tex]
2. Factor out the common factors from each group:
[tex]\[
3a(2m - n) + b(2m - n)
\][/tex]
3. Notice that [tex]\((2m - n)\)[/tex] is a common factor:
[tex]\[
(3a + b)(2m - n)
\][/tex]
Thus, the factorized form is:
[tex]\[
6ma - nb - 3na + 2mb = (3a + b)(2m - n)
\][/tex]
### (d) [tex]\(2ax + by - 2ay - bx\)[/tex]
1. Group the terms to facilitate factoring:
[tex]\[
2ax - 2ay + by - bx = 2a(x - y) + b(y - x)
\][/tex]
2. Rewriting the second term to show the common factor:
[tex]\[
2a(x - y) - b(x - y)
\][/tex]
3. Notice that [tex]\((x - y)\)[/tex] is a common factor (keeping track of the sign change):
[tex]\[
(2a - b)(x - y)
\][/tex]
Thus, the factorized form is:
[tex]\[
2ax + by - 2ay - bx = (2a - b)(x - y)
\][/tex]
In conclusion, the factorized results for each expression are:
(a) [tex]\(2ax + 6ap + bx + 3bp = (2a + b)(3p + x)\)[/tex]
(b) [tex]\(xs - xk - ys + yk = (s - k)(x - y)\)[/tex]
(c) [tex]\(6ma - nb - 3na + 2mb = (3a + b)(2m - n)\)[/tex]
(d) [tex]\(2ax + by - 2ay - bx = (2a - b)(x - y)\)[/tex]
