• question_answer 15)                 Multiply the binomials: (i) (2x + 5) and (4x - 3) (ii) (y - 8) and (3y - 4) (iii) (2.51 - 0.5 m) and (2.51 + 0.5m) (iv) (a + 3b) and (x + 5) (v) (2pq + 3q2) and (3pq - 2q2) (vi) $\left( \frac{3}{4}{{a}^{2}}+3{{b}^{2}} \right)\,$ and $4\left( {{a}^{2}}-\frac{2}{3}{{b}^{2}} \right)$

(i) $(2x+5)$ and $(4x-3)$ $(2x+5)\,\times (4x-3)$ $=(2x)\times (4x-3)+5\times (4x-3)$ $=(2x)\times (4x)-(2x)\times (3)+(5)$$\times (4x)-(5)\times (3)$ $=8{{x}^{2}}-6x+20x-15$ $=8{{x}^{2}}+(20x-6x)-15$                         |Combining like terms $=8{{x}^{2}}+14x-15$ (ii) (y ? 8) and (3y ? 4) $(y-8)\times (3y-4)$ $=y\times (3y-4)-8\times (3y-4)$ $=(y)\times (3y)-(y)\times (4)-(8)$$\times (3y)+8\times 4$ $=3{{y}^{2}}-4y-24y+32$ $=3{{y}^{2}}-28y+32$                  |Combining like terms (iii) $(2.5\,l-0.5m)$ and $(2.5\,l+0.5\,m)$ $(2.5\,l+0.5\,m)\times (2.5\,l+0.5\,m)$ $=(2.5\,l)\times (2.5\,l+0.5\,m)$$-(0.5\,m)\times (2.5l\,+0.5m)$ $=(2.5\,l)\times (2.5\,l)+(2.5\,l)$$\times (0.5\,m)-(0.5\,m)\times (2.5\,l)$$-(0.5\,m)\,\times (0.5\,m)$ $=6.25\,{{l}^{2}}+1.25\,lm-1.25\,ml$$-0.25\,{{m}^{2}}$                                             |Combining like terms $=6.25\,{{l}^{2}}-0.25\,{{m}^{2}}$ (iv) $(a+3b)$ and $(x+5)$ $(a+3b)\times (x+5)$ $=a\times (x+5)+(3b)\,\times (x+5)$ $=(a)\times (x)+(a)\times (5)+(3b)$$\times (x)+(3b)\,\times (5)$ $=ax+5a+3bx+15b$ (v) $(2pq+3{{q}^{2}})$ and $(3pq-2{{q}^{2}})$ $(2pq+3{{q}^{2}})\times (3pq-2{{q}^{2}})$ $=(2pq)\times (3pq-2{{p}^{2}})+(3{{p}^{2}})$$\times (3pq-2{{q}^{2}})$ $=(2pq)\times (3pq)-(2pq)\times (2{{q}^{2}})$ $+(3{{p}^{2}})\times (3pq)-(3{{q}^{2}})\times (2{{q}^{2}})$ $=6{{p}^{2}}{{q}^{2}}-4p{{q}^{3}}+9p{{q}^{3}}-6{{q}^{4}}$ $=6{{p}^{2}}{{q}^{2}}\,+(9p{{q}^{3}}-4p{{q}^{3}})-6{{q}^{4}}$                  |Combining like terms $=6{{p}^{2}}{{q}^{2}}+5p{{q}^{3}}-6{{q}^{4}}$ (vi) $\left( \frac{3}{4}{{a}^{2}}+3{{b}^{2}} \right)$ and $4\left( {{a}^{2}}-\frac{2}{3}{{b}^{2}} \right)$ $\left( \frac{3}{4}{{a}^{2}}+3{{b}^{2}} \right)\times 4\left( {{a}^{2}}-\frac{2}{3}{{b}^{2}} \right)$ $=\left( \frac{3}{4}{{a}^{2}}+3{{b}^{2}} \right)\,\times \left( 4{{a}^{2}}-\frac{8}{3}{{b}^{2}} \right)$ $=\frac{3}{4}{{a}^{2}}\times \left( 4{{a}^{2}}-\frac{8}{3}{{b}^{2}} \right)+3{{b}^{2}}$$\times \,\left( 4{{a}^{2}}-\frac{8}{3}{{b}^{2}} \right)$                 $=\left( \frac{3}{4}{{a}^{2}} \right)\times \,(4{{a}^{2}})-\left( \frac{3}{4}{{a}^{2}} \right)\times \left( \frac{8}{3}{{b}^{2}} \right)$ $+(3{{b}^{2}})\times \,(4{{a}^{2}})-(3{{b}^{2}})\times \left( \frac{8}{3}{{b}^{2}} \right)$ $=3{{a}^{4}}-2{{a}^{2}}{{b}^{2}}+12{{b}^{2}}{{a}^{2}}-8{{b}^{4}}$ $=3{{a}^{4}}+(12{{a}^{2}}{{b}^{2}}-2{{a}^{2}}{{b}^{2}})-8{{b}^{4}}$        |Combining like terms $=3{{a}^{4}}+10{{a}^{2}}{{b}^{2}}-8{{b}^{4}}$.