Answer:
Part 1
Set B
Part 2
We can use the FOIL method to multiply these expressions:
[tex]\textsf{FOIL}: \quad(a+b)(c+d)=ac+ad+bc+bd[/tex]
However, as all the expression are in the format [tex](a+b)^2[/tex], we can use the shortcut: [tex](a+b)^2=a^2+2ab+b^2[/tex]
Part 3
Using the shortcut to find the products:
[tex]\begin{aligned}\implies (x+5)(x+5)& =(x+5)^2\\ & =x^2+2(x)(5)+5^2\\ & =x^2+10x+25\end{aligned}[/tex]
[tex]\begin{aligned}\implies (2x+9)(2x+9) & = (2x+9)^2\\ & = (2x)^2+2(2x)(9)+9^2\\ & = 4x^2+36x+81\end{aligned}[/tex]
[tex]\begin{aligned}\implies (x+1)^2 & = x^2+2(x)(1)+1^2\\ & = x^2+2x+1\end{aligned}[/tex]
Part 4
A new example of a multiplication problem based on the structure of Set B is: [tex](3x+2)(3x+2)[/tex]
[tex]\begin{aligned}\implies (3x+2)(3x+2) & = (3x+2)^2\\ & = (3x)^2+2(3x)(2)+2^2\\ & = 9x^2+12x+4\end{aligned}[/tex]
Part 5
An example of a multiplication problem that would NOT be included in Set B based on its structure is: [tex](x+2)(x-2)[/tex]
We cannot use the derived Set B shortcut (from part 2) to multiply this expression. Instead, we would have to use a different shortcut of "The Difference of Two Squares" to find the product of this example expression.
