Sure, let's multiply the given binomials step by step:
### Part (a):
Multiply [tex]\((x - y)\)[/tex] by [tex]\((x - y)\)[/tex].
1. Use the distributive property (also known as the FOIL method for binomials):
[tex]\[
(x - y)(x - y) = x(x - y) - y(x - y)
\][/tex]
2. Distribute each term:
[tex]\[
= x^2 - xy - yx + y^2
\][/tex]
3. Combine like terms:
[tex]\[
= x^2 - 2xy + y^2
\][/tex]
So, the result of multiplying [tex]\((x - y)\)[/tex] by [tex]\((x - y)\)[/tex] is:
[tex]\[
x^2 - 2xy + y^2
\][/tex]
### Part (c):
Multiply [tex]\(\left(a^2 b^2 + b^2 c^2\right)\)[/tex] by [tex]\((a b - b c)\)[/tex].
1. Distribute each term in the first binomial by each term in the second binomial:
[tex]\[
(a^2 b^2 + b^2 c^2)(a b - b c) = (a^2 b^2)(a b) - (a^2 b^2)(b c) + (b^2 c^2)(a b) - (b^2 c^2)(b c)
\][/tex]
2. Multiply the terms accordingly:
[tex]\[
= a^3 b^3 - a^2 b^3 c + a b^3 c^2 - b^3 c^3
\][/tex]
So, the result of multiplying [tex]\(\left(a^2 b^2 + b^2 c^2\right)\)[/tex] by [tex]\((a b - b c)\)[/tex] is:
[tex]\[
a^3 b^3 - a^2 b^3 c + a b^3 c^2 - b^3 c^3
\][/tex]
Thus, the final answers are:
(a) [tex]\( x^2 - 2xy + y^2 \)[/tex]
(c) [tex]\( a^3 b^3 - a^2 b^3 c + a b^3 c^2 - b^3 c^3 \)[/tex]
