1) Area of larger region = (3x-5)(3x-5)
Area of smaller region = (x+1)(x+1)
Subtract area of smaller region from area of larger region:
[tex] (3x-5)(3x-5) - (x+1)(x+1)[/tex]
⇓
[tex]9x^2 -30x +25 - (x^2+2x+1) = 9x^2 -30x +25 -x^2-2x-1[/tex]
⇓
[tex]8x^2 -32x +24 = 8(x^2-4x+3)[/tex]
⇓
[tex]\boxed{8(x-3)(x-1)}[/tex]
2) Length of shaded region = a³ - b³
Width of shaded region = a² + ab +b²
[tex]Perimeter=2(l + w)=2(a^3-b^3+a^2+ab+b^2)[/tex]
(But [tex]a^3-b^3= (a-b)(a^2+ab+b^2)[/tex] )
So, [tex]2(a^3-b^3+a^2+ab+b^2) = 2[(a-b)(a^2+ab+b^2)+(a^2+ab+b^2)][/tex]
⇓
Final factored form for perimeter = [tex]\boxed{2(a^2+ab+b^2)(a-b+1)}[/tex]
[tex]Area = l\times w= (a^3-b^3)(a^2+ab+b^2)[/tex]
(But [tex]a^3-b^3= (a-b)(a^2+ab+b^2)[/tex] )
So, the final factored form for Area = [tex]\boxed{(a-b)(a^2+ab+b^2)^2}[/tex]
