Question

Simplify: ${(3x+4y+5)}^2-{(x+5y-4)}^2$

Answer 1

Usig the identity: $(a+b+c{)}^2=a^2+b^2+c^2+2ab+2bc+2ca$.

GIven: ${(3x+4y+5)}^2-{(x+5y-4)}^2$ we apply the above identity as follows:

${(3x+4y+5)}^2-{(x+5y-4)}^2=[{\left(3x\right)}^2+{\left(4y\right)}^2+{\left(5\right)}^2+(2\times 3x\times 4y)+(2\times 4y\times 5)+(2\times 5\times 3x)]$

$-[{\left(x\right)}^2+{\left(5y\right)}^2+{(-4)}^2+(2\times x\times 5y)+(2\times 5y\times -4)+(2\times -4\times x)]=(9x^2+16y^2+25+24xy+40y+30x)$

$-(x^2+25y^2+16+10xy-40y-8x)=9x^2+16y^2+25+24xy+40y+30x-x^2-25y^2-16-10xy+40y+8x$

$=8x^2-9y^2+9+14xy+80y+38x$

Hence, ${(3x+4y+5)}^2-{(x+5y-4)}^2=8x^2-9y^2+9+14xy+80y+38x$.