\displaystyle \textbf{Question 1: }\text{Multiply:}
\displaystyle \text{(i) }8x^2\text{ by }-5x
\displaystyle \text{(ii) }\frac{2}{3}ab\text{ by }-6a^2b
\displaystyle \text{(iii) }7x^2y^3\text{ by }-4x^3y
\displaystyle \text{(iv) }\left(-\frac{5}{9}ax^2\right) \text{ by }\left(-\frac{3}{5}bxa^2\right)
\displaystyle \text{(v) }\left(-\frac{2}{3}a^6b^5\right) \text{ by }\left(-\frac{9}{4}a^3b^3\right)
\displaystyle \text{(vi) }\left(-\frac{5}{8}p^2q\right) \text{ by }\left(\frac{16}{25}(pq)^2\right)
\displaystyle \text{Answer:}

\displaystyle \text{(i) }8x^2\times(-5x) =(8\times-5)x^{2+1}=-40x^3.

\displaystyle \text{(ii) }\frac{2}{3}ab\times(-6a^2b) =\left(\frac{2}{3}\times-6\right)a^{1+2}b^{1+1}
\displaystyle =-4a^3b^2.

\displaystyle \text{(iii) }7x^2y^3\times(-4x^3y) =(7\times-4)x^{2+3}y^{3+1}
\displaystyle =-28x^5y^4.

\displaystyle \text{(iv) }\left(-\frac{5}{9}ax^2\right) \left(-\frac{3}{5}bxa^2\right)
\displaystyle =\left(-\frac{5}{9}\times-\frac{3}{5}\right) a^{1+2}x^{2+1}b=\frac{1}{3}a^3bx^3.

\displaystyle \text{(v) }\left(-\frac{2}{3}a^6b^5\right) \left(-\frac{9}{4}a^3b^3\right)
\displaystyle =\left(-\frac{2}{3}\times-\frac{9}{4}\right) a^{6+3}b^{5+3}=\frac{3}{2}a^9b^8.

\displaystyle \text{(vi) }\left(-\frac{5}{8}p^2q\right) \left(\frac{16}{25}(pq)^2\right)
\displaystyle =\left(-\frac{5}{8}\times\frac{16}{25}\right) p^{2+2}q^{1+2}=-\frac{2}{5}p^4q^3.
\displaystyle \\

\displaystyle \textbf{Question 2: }\text{Multiply:}
\displaystyle \text{(i) }(2x^2-5x+6)\text{ by }-3x
\displaystyle \text{(ii) }(3-2y-y^2)\text{ by }-4xy
\displaystyle \text{(iii) }(3x^2y-2xy^2+5xy-6)\text{ by }4xy
\displaystyle \text{(iv) }(7a^3-5ab^2-2b^3+3ab+2a-5) \text{ by }-3ab^2
\displaystyle \text{Answer:}

\displaystyle \text{(i) }(2x^2-5x+6)(-3x)
\displaystyle =(2x^2)(-3x)+(-5x)(-3x)+6(-3x)
\displaystyle =-6x^3+15x^2-18x.

\displaystyle \text{(ii) }(3-2y-y^2)(-4xy)
\displaystyle =3(-4xy)+(-2y)(-4xy)+(-y^2)(-4xy)
\displaystyle =-12xy+8xy^2+4xy^3.

\displaystyle \text{(iii) }(3x^2y-2xy^2+5xy-6)(4xy)
\displaystyle =(3x^2y)(4xy)+(-2xy^2)(4xy)+(5xy)(4xy)-6(4xy)
\displaystyle =12x^3y^2-8x^2y^3+20x^2y^2-24xy.

\displaystyle \text{(iv) }(7a^3-5ab^2-2b^3+3ab+2a-5)(-3ab^2)
\displaystyle =(7a^3)(-3ab^2)+(-5ab^2)(-3ab^2) +(-2b^3)(-3ab^2)
\displaystyle \quad +(3ab)(-3ab^2)+(2a)(-3ab^2) +(-5)(-3ab^2)
\displaystyle =-21a^4b^2+15a^2b^4+6ab^5-9a^2b^3 -6a^2b^2+15ab^2.
\displaystyle \\

\displaystyle \textbf{Question 3: }\text{Multiply:}
\displaystyle \text{(i) }(a+5)\text{ by }(a+4)\qquad \text{(ii) }(x-3)\text{ by }(x+8)
\displaystyle \text{(iii) }(y-4)\text{ by }(y-6)\qquad \text{(iv) }(z+1)\text{ by }(z-7)
\displaystyle \text{(v) }(3a-2b)\text{ by }(2a+3b)\qquad \text{(vi) }(5x+4y)\text{ by }(2x-3y)
\displaystyle \text{Answer:}

\displaystyle \text{(i) }(a+5)(a+4) =a^2+4a+5a+20=a^2+9a+20.

\displaystyle \text{(ii) }(x-3)(x+8) =x^2+8x-3x-24=x^2+5x-24.

\displaystyle \text{(iii) }(y-4)(y-6) =y^2-6y-4y+24=y^2-10y+24.

\displaystyle \text{(iv) }(z+1)(z-7) =z^2-7z+z-7=z^2-6z-7.

\displaystyle \text{(v) }(3a-2b)(2a+3b)
\displaystyle =6a^2+9ab-4ab-6b^2 =6a^2+5ab-6b^2.

\displaystyle \text{(vi) }(5x+4y)(2x-3y)
\displaystyle =10x^2-15xy+8xy-12y^2 =10x^2-7xy-12y^2.
\displaystyle \\

\displaystyle \textbf{Question 4: }\text{Multiply:}
\displaystyle \text{(i) }(4x^2+3x-5)\text{ by }(2x+3)
\displaystyle \text{(ii) }(3-2x+5x^2)\text{ by }(5x-4)
\displaystyle \text{(iii) }(x^3-5x+3)\text{ by }(2x+9)
\displaystyle \text{(iv) }(4x^2+xy+9y^2)\text{ by }(2x-3y)
\displaystyle \text{Answer:}

\displaystyle \text{(i) }(4x^2+3x-5)(2x+3)
\displaystyle =8x^3+6x^2-10x+12x^2+9x-15
\displaystyle =8x^3+18x^2-x-15.

\displaystyle \text{(ii) }(3-2x+5x^2)(5x-4)
\displaystyle =15x-10x^2+25x^3-12+8x-20x^2
\displaystyle =25x^3-30x^2+23x-12.

\displaystyle \text{(iii) }(x^3-5x+3)(2x+9)
\displaystyle =2x^4-10x^2+6x+9x^3-45x+27
\displaystyle =2x^4+9x^3-10x^2-39x+27.

\displaystyle \text{(iv) }(4x^2+xy+9y^2)(2x-3y)
\displaystyle =8x^3+2x^2y+18xy^2-12x^2y-3xy^2-27y^3
\displaystyle =8x^3-10x^2y+15xy^2-27y^3.
\displaystyle \\

\displaystyle \textbf{Question 5: }\text{Multiply:}
\displaystyle \text{(i) }(4x^2-4x+1)\text{ by }(2x^2+x-2)
\displaystyle \text{(ii) }(3x^2+4x-5)\text{ by }(4x^2-7x+2)
\displaystyle \text{(iii) }(4x^2+24xy+3y^2)\text{ by }(x^2-xy+y^2)
\displaystyle \text{(iv) }(6x^3-5x^2+4x+1)\text{ by }(x^2+7x-1)
\displaystyle \text{(v) }(8x^4-3x^2+9x-8)\text{ by }(2x^2-5x+3)
\displaystyle \text{(vi) }(3x^5-7x^3+2x^2-x+4)\text{ by}
\displaystyle (x^3-2x^2+3x-1)
\displaystyle \text{Answer:}

\displaystyle \text{(i) }(4x^2-4x+1)(2x^2+x-2)
\displaystyle =8x^4+4x^3-8x^2-8x^3-4x^2+8x+2x^2+x-2
\displaystyle =8x^4-4x^3-10x^2+9x-2.

\displaystyle \text{(ii) }(3x^2+4x-5)(4x^2-7x+2)
\displaystyle =12x^4-21x^3+6x^2+16x^3-28x^2+8x
\displaystyle \quad -20x^2+35x-10
\displaystyle =12x^4-5x^3-42x^2+43x-10.

\displaystyle \text{(iii) }(4x^2+24xy+3y^2)(x^2-xy+y^2)
\displaystyle =4x^4+24x^3y+3x^2y^2-4x^3y-24x^2y^2-3xy^3
\displaystyle \quad +4x^2y^2+24xy^3+3y^4
\displaystyle =4x^4+20x^3y-17x^2y^2+21xy^3+3y^4.

\displaystyle \text{(iv) }(6x^3-5x^2+4x+1)(x^2+7x-1)
\displaystyle =6x^5-5x^4+4x^3+x^2+42x^4-35x^3+28x^2+7x
\displaystyle \quad -6x^3+5x^2-4x-1
\displaystyle =6x^5+37x^4-37x^3+34x^2+3x-1.

\displaystyle \text{(v) }(8x^4-3x^2+9x-8)(2x^2-5x+3)
\displaystyle =16x^6-6x^4+18x^3-16x^2-40x^5+15x^3-45x^2+40x
\displaystyle \quad +24x^4-9x^2+27x-24
\displaystyle =16x^6-40x^5+18x^4+33x^3-70x^2+67x-24.

\displaystyle \text{(vi) }(3x^5-7x^3+2x^2-x+4)(x^3-2x^2+3x-1)
\displaystyle =3x^8-7x^6+2x^5-x^4+4x^3-6x^7+14x^5-4x^4
\displaystyle \quad +2x^3+9x^6-21x^4+6x^3-3x^2-3x^5+7x^3
\displaystyle \quad -2x^2+x+12x-8x^2+12x-4
\displaystyle =3x^8-6x^7+2x^6+13x^5-26x^4+19x^3-13x^2+13x-4.
\displaystyle \\

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