• # question_answer 14)                 (a) Add: $p(p-q),\,q(q-r)$ and $r(r-p)$                 (b) Add: $2x(z-x-y)$ and $2y(z-y-x)$                 (c) Subtract: $3l(l-4m+5n)$ from $(10n-3m+2l)$ (d) Subtract: $3a(a+b+c)\,-2b(a-b+c)$ from $4c(-a+b+c)$.

(a) First expression $=p(p-q)\,=p\times p-p\times q$ $={{p}^{2}}-pq$ Second expression                 $=q(q-r)$ $=q\times q-q\times r$ $={{q}^{2}}-qr$                 Third expression $=r(r-p)$ $=r\times r-r\times p$ $={{r}^{2}}-rp$                 Adding the three expressions,
 ${{p}^{2}}-pq$ + $+{{q}^{2}}-qr$ + $+{{r}^{2}}-rp$ ${{p}^{2}}-pq+{{q}^{2}}-qr+{{r}^{2}}-rp$
(b) First expression $=2x(z-x-y)$ $=(2x)\times (z)-(2x)\,\times (x)-(2x)\times (y)$ $=2xz-2{{x}^{2}}-2xy$ Second expression $=2y(z-y-x)$ $=(2y)\times (z)-(2y)\times (y)-(2y)\times (x)$ $=2yz-2{{y}^{2}}-2yx$ Adding the two expressions,  $2xz-2{{x}^{2}}-2xy$ + $-2yx+2yz-2{{y}^{2}}$ $2xz-2{{x}^{2}}-4xy+2yz-2{{y}^{2}}$
(c) First expression $=3l\,(l-4m+5n)$ $=(3l)\times (l)-(3l)\times (4m)+(3l)\times (5n)$ $=3{{l}^{2}}-12lm+15\ln$ Second expression $=4l\,(10n-3m+2l)$ $=(4l)\times (10n)-(4l)\times (3m)+(4l)\times (2l)$ $=40\,\ln -12lm\,+8{{l}^{2}}$ Subtraction $\to$  $40\ln -12lm+8{{l}^{2}}$ $15\ln \,-12lm\,+3{{l}^{2}}$ ? + ? $25\ln$ $+5{{l}^{2}}$
(d) First expression $=3a(a+b+c)\,-2b(a-b+c)$ $=(3a)\times (a)+(3a)\times (b)+(3a)$$\times (c)-(2b)\times (a)+(2b)\times (b)$$-(2b)\times (c)$ $=3{{a}^{2}}+3ab+3ac-2ab+2{{b}^{2}}-2bc$ $=3{{a}^{2}}+2{{b}^{2}}+3ab-2ab-2bc+3ac$ $=3{{a}^{2}}+2{{b}^{2}}+ab-2bc+3ac$ Second expression $=4c(-a+b+c)$ $=4c\times (-a)\,+4c\times b+4c\times c$ $=-4ac+4bc+4{{c}^{2}}$ Subtraction $\to$  $-4ac+4bc+4{{c}^{2}}$ $3{{a}^{2}}+2{{b}^{2}}+ab+3ac-2bc$ ? ? ? ? + $-3{{a}^{2}}-2{{b}^{2}}-ab-7ac+6bc+4{{c}^{2}}$