Answer:
(i) \[{{a}^{2}}+8a+16\] \[{{a}^{2}}+8a+16\]\[={{(a)}^{2}}+2(a)(4)+{{(4)}^{2}}\] \[={{(a+4)}^{2}}\] |Applying Identity I (ii) \[{{p}^{2}}-10p+25\] \[{{p}^{2}}-10p+25\] \[={{(p)}^{2}}-2(p)\,(5)\,+{{(5)}^{2}}\] \[={{(p-5)}^{2}}\] |Using Identity II (iii) \[25{{m}^{2}}+30m+9\] \[25{{m}^{2}}+30m+9\] \[={{(5m)}^{2}}\,+2(5m)\,(3)+{{(3)}^{2}}\] \[={{(5m+3)}^{2}}\] |Applying Identity I (iv) \[49{{y}^{2}}+84yz+36{{z}^{2}}\] \[49{{y}^{2}}+84yz+36{{z}^{2}}\] \[={{(7y)}^{2}}+2(7y)\,(6z)\,+{{(6z)}^{2}}\] \[={{(7y+6z)}^{2}}\] |Using Identity I (v) \[4{{x}^{2}}-8x+4\] \[4{{x}^{2}}-8x+4\] \[=4({{x}^{2}}-2x+1)\] \[=4[{{(x)}^{2}}-2(x)(1)+{{(1)}^{2}}]\] \[=4{{(x-1)}^{2}}\] |Applying Identity II (vi) \[121{{b}^{2}}-88bc+16{{c}^{2}}\] \[121{{b}^{2}}-88bc+16{{c}^{2}}\] \[={{(11b)}^{2}}\,-2(11b)\,(4c)\,+{{(4c)}^{2}}\] \[={{(11b-4c)}^{2}}\] |Using Identity II (vii) \[{{(l+m)}^{2}}-4lm\] \[{{(l+m)}^{2}}-4lm\] \[=({{l}^{2}}+2lm+{{m}^{2}})\,-4lm\] |Using Identity I \[={{l}^{2}}+(2lm-4lm)+{{m}^{2}}\] |Combining the like terms \[={{l}^{2}}-2lm+{{m}^{2}}\] \[={{(l)}^{2}}-2(l)(m)+{{(m)}^{2}}\] \[={{(l-m)}^{2}}\] |Applying Identity II (viii) \[{{a}^{4}}+2{{a}^{2}}{{b}^{2}}+{{b}^{4}}\] \[{{a}^{4}}+2{{a}^{2}}{{b}^{2}}+{{b}^{4}}\] \[={{({{a}^{2}})}^{2}}+2({{a}^{2}})\,({{b}^{2}})+{{({{b}^{2}})}^{2}}\] \[={{({{a}^{2}}+{{b}^{2}})}^{2}}\] |Using Identity I
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