Question 1139327
a)

 {{{(a+b+c)(ab+bc+ca)-abc= (a+b)(b+c)(c+a)}}}

manipulate left side

 {{{(a+b+c)(ab+bc+ca)-abc}}}

={{{a^2b+abc+a^2c   +ab^2+b^2c+bca    +cross(abc)+bc^2+c^2a-cross(abc)}}}

={{{a^2b+abc+a^2c   +ab^2+b^2c+bca   +bc^2+c^2a }}} ...group

={{{(a^2b+a^2c)+(abc +ab^2)  +(c^2a+ bca   ) +  (b^2c+bc^2) }}} 
 
={{{a^2(b+c)+ab(b+c)  +ca(c+ b   ) + bc (b+c)  }}}

={{{(a^2+ab  +ca + bc) (b+c) }}} 

={{{((a^2+ab)  +(ca + bc)) (b+c) }}} 

={{{(a(a+b)  +c(a + b)) (b+c) }}} 

={{{(a +c)(a + b) (b+c)  }}}


b.)

{{{(ax+by)^2 + (ay-bx)^2 +c^2(x^2 +y^2)= (x^2 +y^2)(a^2 +b^2 +c^2) }}}

manipulate left side:

{{{(ax+by)^2 + (ay-bx)^2 +c^2(x^2 +y^2)}}}

={{{a^2*x^2 +cross( 2 abxy) + b^2*y^2 +a^2*y^2 - cross( 2 a y b x) + b^2*x^2 +c^2*x^2 +c^2*y^2}}}

={{{a^2 *x^2 + b^2* x^2 +c^2*x^2  +a^2* y^2+ b^2* y^2 +c^2*y^2}}}

={{{x^2 (a^2  + b^2 +c^2) +y^2(a^2 + b^2  +c^2)}}}

={{{(x^2  +y^2)(a^2 + b^2  +c^2)}}}