Question 1: Find the following products;

$i) \left(x+5\right)\left(x+7\right)$

$=\ x^2+5x+7x+35$

$={\ x}^2+12x+35$

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$ii) \left(b+2\right)\left(b+9\right)$

$={\ b}^2+2b+9b+18$

$={\ b}^2+11b+18$

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$iii) \left(c+2\right)\left(c+\frac{3}{5}\right)$

$={\ c}^2+2c+\frac{3}{5}c+\frac{6}{5}$

$={\ c}^2+\left(2+\frac{3}{5}\right)c+\frac{6}{5}$

$=\ c^2+\frac{13}{5}c+\frac{6}{5}$

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$iv) \left(t+\frac{4}{3}\right)\left(t+\frac{1}{3}\right)$

$={\ t}^2+\frac{4}{3}t+\frac{1}{3}t+\frac{4}{9}$

$={\ t}^2+\frac{5}{3}t+\frac{4}{9}$

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Question 2:  Find the following products;

$i) \left(y+8\right)\left(y-4\right)$

$=\ y^2+8y-4y-32$

$={\ y}^2+4y-32$

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$ii) \left(z+6\right)\left(z-11\right)$

$={\ z}^2+6z-11z-66$

$= {\ z}^2-5z-66$

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$iii) \left(c-5\right)\left(c+1\right)$

$={\ c}^2-5c+c-5$

$=\ c^2-4c-5$

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$iv) \left(b-13\right)\left(b+10\right)$

$={\ b}^2-13b+10b-130$

$=b^2-3b-130$

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Question 3: Find the following products;

$i) \left(x-3\right)\left(x-6\right)$

$={\ x}^2-3x-6x+18$

$={\ x}^2-9x+18$

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$ii) \left(z-11\right)\left(z-4\right)$

$=\ z^2-15z+44$

$= \ z^2-11z-4z+44$

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$iii) \left(b-6\right)\left(b-8\right)$

$=\ b^2-6b-8b+48$

$={\ b}^2-14b+48$

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$iv) \left(a-\frac{3}{5}\right)\left(a-\frac{1}{3}\right)$

$={\ a}^2-\frac{3}{5}a-\frac{1}{3}a+\frac{1}{5}$

$= a^2-\frac{14}{15}a+\frac{1}{5}$

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Question 4:  Explain the following;

$i) \left(x-\frac{1}{2}\right)\left(x+\frac{3}{2}\right)$

$={\ x}^2-\frac{1}{2}x+\frac{3}{2}x-\frac{3}{4}$

$={\ x}^2+x-\frac{3}{4}$

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$ii) \left(p-3\right)\left(p+\frac{1}{2}\right)$

$={\ p}^2-3p-\frac{1}{2}p-\frac{3}{2}$

$={\ p}^2-\frac{7}{2}p-\frac{3}{2}$

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Question 5:  Explain the following;

$i) \left(4p+3\right)\left(4p+7\right)$

$={\ 16p}^2+12p+28p+21$

$={\ 16p}^2+40p+21$

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$ii) \left(9c+4\right)\left(9c-2\right)$

$=\ 81c^2+36c-18c-8$

$=\ 81c^2+18c-8$

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$ii) \left(3a-8\right)\left(3a+2\right)$

$=\ {9a}^2-24a+6a-16$

$={\ 9a}^2-18a-16$

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$iv) \left(5x-2\right)\left(5x-7\right)$

$=\ {25x}^2-10x-35x+14$

$={\ 25x}^2-45x+14$

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Question 6:  Find the following products;

$i) \left(x^2+3\right)\left(x^2+6\right)$

$={\ x}^4+{3x}^2+6x^2+18$

$= x^4+9x^2+18$

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$ii) \left(y^2-1\right)\left(y^2+4\right)$

$=\ y^4-y^2+{4y}^2-4$

$={\ y}^4+3y^2-4$

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$iii) \left(z^2+3\right)\left(z^2-7\right)$

$={\ z}^4+3z^2-7z^2-21$

$={\ z}^4-4z^2-21$

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$iv) \left(t^2-2\right)\left(t^2-5\right)$

$={\ t}^4-2t^2-5t^2+10$

$=\ t^4-7t^2+10$

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Question 7:  Find the following products;

$i) \left(ab-2\right)\left(ab+4\right)$

$={\ a}^2b^2-2ab++4ab-8$

$={\ a}^2b^2+2ab-8$

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$ii) \left(2+xy\right)\left(3-xy\right)$

$=\ {6+3xy-2xy-x}^2y^2$

$={\ 6+xy-x}^2y^2$

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Question 8:  Find the following products;

$i) \left(3x+4y\right)\left(4x+3y\right)$

$=\ 12x^2+16xy+9xy+12y^2$

$=\ 12x^2+25xy+12y^2$

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$ii) \left(4a-5b\right)\left(3a+2b\right)$

$=\ 12a^2-15ab+8ab-10b^2$

$=\ 12a^2-7ab-10b^2$

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$iii) \left(2y+z\right)\left(7z-3y\right)$

$=\ 14yz+7z^2-6y^2-3yz$

$=\ 7z^2+11yz-6y^2$

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$iv) \left(2m-4n\right)\left(4m-3n\right)$

$=\ 8m^2-16mn-9mn-12n^2$

$={\ m}^2-25mn-12n^2$

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Question 9:  Find the following products;

$i) \left(2pq+0.1mn\right)\left(0.2pq+3mn\right)$

$=\ 0.4p^2q^2+0.02pqmn+6pqmn+0.3m^2n^2$

$=\ 0.4p^2q^2+6.02pqmn+0.3m^2n^2$

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$ii) \left(\frac{4m}{p}-\frac{0.2n}{q}\right)\left(\frac{3m}{p}+\frac{0.5}{q}\right)$

$= \frac{12m^2}{p^2}-\frac{0.6mn}{pq}+\frac{2mn}{pq}-\frac{0.1n^2}{q^2}$

$= \ \frac{12m^2}{p^2}+\frac{1.4mn}{pq}-\frac{0.1n^2}{q^2}$

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$iii) \left(\frac{3}{4}x-2pq\right)\left(3x-\frac{4}{5}pq\right)$

$=\ \frac{9}{4}x^2-6pqx-\frac{3}{5}pqx-\frac{8}{5}p^2q^2$

$=\ \frac{9}{4}x^2-\frac{33}{5}pqx-\frac{8}{5}p^2q^2$

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Question 10:  Find the following products;

$i) \left(2a^2+3b^2\right)\left(3a^2+2b^2\right)$

$=\ 4a^4+9a^2b^2+4a^2b^2+9a^2b^2$

$=\ 4a^4+13a^2b^2+9a^2b^2$

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$ii) \left(x^2+5y^2\right)\left(y^2-2x^2\right)$

$={\ x}^2y^2+5y^4-2x^4-10x^2y^2$

$=\ 5y^4-9x^2y^2-2x^4$

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$iii) \left(3c^2-4d^2\right)\left(4c^2-3d^2\right)$

$=\ 12c^4-16c^2d^2-9c^2d^2+12d^4$

$=\ 12c^4-25c^2d^2+12d^4$

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