Answer:
(i) \[{{p}^{2}}+6p+8\] \[{{p}^{2}}+6p+8\] \[={{p}^{2}}+6p+9-1\] \[=\{{{(p)}^{2}}+2\,(p)\,(3)\,+{{(3)}^{2}}\}\,-{{(1)}^{2}}\] \[={{(p+3)}^{2}}-{{(1)}^{2}}\] |Using Identity I \[=(p+3-1)\,(p+3+1)\] |Using Identity III \[=(p+2)\,(p\,+4)\] (ii) \[{{q}^{2}}-10q+21\] \[{{q}^{2}}-10q+21\] \[={{q}^{2}}-10q+25-4\] \[=\{{{(p)}^{2}}-2(q)\,(5)\,+{{(5)}^{2}}\}-4\] \[={{(q-5)}^{2}}\,-{{(2)}^{2}}\] |Using Identity II \[=(q-5-2)\,(q-5+2)\] |Using Identity III \[=(q-7)\,(q-3)\] (iii) \[{{p}^{2}}+6p-16\] \[{{p}^{2}}+6p-16\] \[={{p}^{2}}+6p+9-25\] \[={{(p)}^{2}}+2(p)\,(3)+{{(3)}^{2}}-{{(5)}^{2}}\] \[={{(p+3)}^{2}}-{{(5)}^{2}}\] |Using Identity I \[=(p+3-5)\,(p+3+5)\] |Applying Identity III \[=(p-2)\,(p+8)\]
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