\displaystyle \textbf{Question 1: }\text{Factorize } 5xy+20x-9y-36.
\displaystyle \text{Answer:}
\displaystyle 5xy+20x-9y-36
\displaystyle =5x(y+4)-9(y+4)
\displaystyle =(y+4)(5x-9).
\displaystyle \\

\displaystyle \textbf{Question 2: }\text{Factorize } 5ab-10b-7a+14.
\displaystyle \text{Answer:}
\displaystyle 5ab-10b-7a+14
\displaystyle =5b(a-2)-7(a-2)
\displaystyle =(a-2)(5b-7).
\displaystyle \\

\displaystyle \textbf{Question 3: }\text{Factorize } 9xy+7y-9y^2-7x.
\displaystyle \text{Answer:}
\displaystyle 9xy+7y-9y^2-7x
\displaystyle =(9xy-9y^2)+(7y-7x)
\displaystyle =9y(x-y)-7(x-y)
\displaystyle =(x-y)(9y-7).
\displaystyle \\

\displaystyle \textbf{Question 4: }\text{Factorize } 3ax-6ay-8by+4bx.
\displaystyle \text{Answer:}
\displaystyle 3ax-6ay-8by+4bx
\displaystyle =3a(x-2y)+4b(x-2y)
\displaystyle =(x-2y)(3a+4b).
\displaystyle \\

\displaystyle \textbf{Question 5: }\text{Factorize } 4x^2-10xy-6xz+15yz.
\displaystyle \text{Answer:}
\displaystyle 4x^2-10xy-6xz+15yz
\displaystyle =2x(2x-5y)-3z(2x-5y)
\displaystyle =(2x-5y)(2x-3z).
\displaystyle \\

\displaystyle \textbf{Question 6: }\text{Factorize } 3ab-6bc-3ax+6cx.
\displaystyle \text{Answer:}
\displaystyle 3ab-6bc-3ax+6cx
\displaystyle =3b(a-2c)-3x(a-2c)
\displaystyle =3(a-2c)(b-x).
\displaystyle \\

\displaystyle \textbf{Question 7: }\text{Factorize } a^2+ab(1+b)+b^3.
\displaystyle \text{Answer:}
\displaystyle a^2+ab(1+b)+b^3
\displaystyle =a^2+ab+ab^2+b^3
\displaystyle =a(a+b)+b^2(a+b)
\displaystyle =(a+b)(a+b^2).
\displaystyle \\

\displaystyle \textbf{Question 8: }\text{Factorize } xy^2+(x-1)y-1.
\displaystyle \text{Answer:}
\displaystyle xy^2+(x-1)y-1
\displaystyle =xy^2+xy-y-1
\displaystyle =xy(y+1)-(y+1)
\displaystyle =(y+1)(xy-1).
\displaystyle \\

\displaystyle \textbf{Question 9: }\text{Factorize } b^2-ab(1-a)-a^3.
\displaystyle \text{Answer:}
\displaystyle b^2-ab(1-a)-a^3
\displaystyle =b^2-ab+a^2b-a^3
\displaystyle =b(b-a)+a^2(b-a)
\displaystyle =(b-a)(b+a^2).
\displaystyle \\

\displaystyle \textbf{Question 10: }\text{Factorize } y^2-y(2b+a)+2ab.
\displaystyle \text{Answer:}
\displaystyle y^2-y(2b+a)+2ab
\displaystyle =y^2-2by-ay+2ab
\displaystyle =y(y-2b)-a(y-2b)
\displaystyle =(y-2b)(y-a).
\displaystyle \\

\displaystyle \textbf{Question 11: }\text{Factorize } x-2-(x-2)^2+ax-2a.
\displaystyle \text{Answer:}
\displaystyle x-2-(x-2)^2+ax-2a
\displaystyle =(x-2)-(x-2)^2+a(x-2)
\displaystyle =(x-2)\{1-(x-2)+a\}
\displaystyle =(x-2)(3-x+a).
\displaystyle \\

\displaystyle \textbf{Question 12: }\text{Factorize } x-1-(x-1)^2+4-4x.
\displaystyle \text{Answer:}
\displaystyle x-1-(x-1)^2+4-4x
\displaystyle =(x-1)-(x-1)^2-4(x-1)
\displaystyle =(x-1)\{1-(x-1)-4\}
\displaystyle =(x-1)(-x-2)
\displaystyle =(1-x)(x+2).
\displaystyle \\

\displaystyle \textbf{Question 13: }\text{Factorize } b(c-d)^2-a(d-c)+5c-5d.
\displaystyle \text{Answer:}
\displaystyle d-c=-(c-d).
\displaystyle \therefore b(c-d)^2-a(d-c)+5c-5d
\displaystyle =b(c-d)^2+a(c-d)+5(c-d)
\displaystyle =(c-d)\{b(c-d)+a+5\}
\displaystyle =(c-d)(bc-bd+a+5).
\displaystyle \\

\displaystyle \textbf{Question 14: }\text{Factorize } ab(c^2+b^2)-a^2cd-b^2cd.
\displaystyle \text{Answer:}
\displaystyle ab(c^2+b^2)-a^2cd-b^2cd
\displaystyle =abc^2+ab^3-a^2cd-b^2cd.
\displaystyle \text{This expression does not factorize by the grouping shown.}
\displaystyle \\

\displaystyle \textbf{Question 15: }\text{Factorize } ab(x^2+y^2)+xy(a^2+b^2).
\displaystyle \text{Answer:}
\displaystyle ab(x^2+y^2)+xy(a^2+b^2)
\displaystyle =abx^2+aby^2+a^2xy+b^2xy
\displaystyle =ax(bx+ay)+by(ay+bx)
\displaystyle =(bx+ay)(ax+by).
\displaystyle \\

\displaystyle \textbf{Question 16: }\text{Factorize } ab(x^2+y^2)-xy(a^2+b^2).
\displaystyle \text{Answer:}
\displaystyle ab(x^2+y^2)-xy(a^2+b^2)
\displaystyle =abx^2+aby^2-a^2xy-b^2xy
\displaystyle =ax(bx-ay)+by(ay-bx)
\displaystyle =ax(bx-ay)-by(bx-ay)
\displaystyle =(bx-ay)(ax-by).
\displaystyle \\

\displaystyle \textbf{Question 17: }\text{Factorize } (ax+by)^2+(bx-ay)^2.
\displaystyle \text{Answer:}
\displaystyle (ax+by)^2+(bx-ay)^2
\displaystyle =a^2x^2+2abxy+b^2y^2 +b^2x^2-2abxy+a^2y^2
\displaystyle =a^2x^2+b^2y^2+b^2x^2+a^2y^2
\displaystyle =a^2(x^2+y^2)+b^2(x^2+y^2)
\displaystyle =(x^2+y^2)(a^2+b^2).
\displaystyle \\

\displaystyle \textbf{Question 18: }\text{Factorize } a(a+b+c)-bc.
\displaystyle \text{Answer:}
\displaystyle a(a+b+c)-bc
\displaystyle =a^2+ab+ac-bc.
\displaystyle \text{This expression does not factorize by the grouping shown.}
\displaystyle \\

\displaystyle \textbf{Question 19: }\text{Factorize } a(a-2b-c)+2bc.
\displaystyle \text{Answer:}
\displaystyle a(a-2b-c)+2bc
\displaystyle =a^2-2ab-ac+2bc
\displaystyle =a(a-c)-2b(a-c)
\displaystyle =(a-c)(a-2b).
\displaystyle \\

\displaystyle \textbf{Question 20: }\text{Factorize } 3x^5-6x^4-2x^3+4x^2+x-2.
\displaystyle \text{Answer:}
\displaystyle 3x^5-6x^4-2x^3+4x^2+x-2
\displaystyle =3x^4(x-2)-2x^2(x-2)+(x-2)
\displaystyle =(x-2)(3x^4-2x^2+1).
\displaystyle \\

\displaystyle \textbf{Question 21: }\text{Factorize } 2a+2b-3cx-3bx+2c-3ax.
\displaystyle \text{Answer:}
\displaystyle 2a+2b-3cx-3bx+2c-3ax
\displaystyle =(2a-3ax)+(2b-3bx)+(2c-3cx)
\displaystyle =a(2-3x)+b(2-3x)+c(2-3x)
\displaystyle =(2-3x)(a+b+c).
\displaystyle \\

\displaystyle \textbf{Question 22: }\text{Factorize } a^2b^2+b^2+a^2+1.
\displaystyle \text{Answer:}
\displaystyle a^2b^2+b^2+a^2+1
\displaystyle =b^2(a^2+1)+(a^2+1)
\displaystyle =(a^2+1)(b^2+1).
\displaystyle \\

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