81[x+1]raise to power two+90[x+1][y+2]+25[y+2]raise to power two

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Solution

$81 {(x + 1)}^{2} + 90 (x + 1) (y + 2) + 25 {(y + 2)}^{2} = {[9 (x + 1)]}^{2} + 2 \{9 (x + 1)\} \{5 (y + 2)\} + {[5 (y + 2)]}^{2} = {[9 (x + 1) + 5 (y + 2)]}^{2} [as, a^{2} + 2 a b + b^{2} = {(a + b)}^{2}] = {(9 x + 9 + 5 y + 10)}^{2} = {(9 x + 5 y + 19)}^{2} = (9 x + 5 y + 19) (9 x + 5 y + 19)$

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81[x+1]raise to power two+90[x+1][y+2]+25[y+2]raise to power two

81x+12 + 90x+1y+2 + 25y+22=9x+12 + 29x+1 5y+2 + 5y+22=9x+1 + 5y+22 as, a2 + 2ab + b2 = a+b2=9x + 9 + 5y + 102=9x + 5y + 192=9x + 5y + 19 9x + 5y + 19

$81 {(x + 1)}^{2} + 90 (x + 1) (y + 2) + 25 {(y + 2)}^{2} = {[9 (x + 1)]}^{2} + 2 \{9 (x + 1)\} \{5 (y + 2)\} + {[5 (y + 2)]}^{2} = {[9 (x + 1) + 5 (y + 2)]}^{2} [as, a^{2} + 2 a b + b^{2} = {(a + b)}^{2}] = {(9 x + 9 + 5 y + 10)}^{2} = {(9 x + 5 y + 19)}^{2} = (9 x + 5 y + 19) (9 x + 5 y + 19)$