Prove that `(a+b+c)^3 - a^3-b^3-c^3 = 3(a+b)(b+c)(c+a).`

Explanation

Given `p(x)=x^(4)-2x^(3)+3x^(2)-ax+3a-7`
when we divide p(x) by x+1 ,then we get the the reamainder p(-1)
Now , `P(-1)=(-1)^(4)-2(-1)^(3)+3(-1)^(3)+3(-1)^(2)-a(-1)+3a-7`
`=1+2+3+a+3a-7=4a-1`
According to the question `p(-1) =19`
`implies 4a-1=19`
`implies 4a-1=19`
` implies 4a=20`
`therefore a=5`
`therefore ` Required polynomial `=x^(4)-2x^(3)+3x^(2)-5x+3(5)-7`
`=x^(4) -2x^(3) +3x^(2)-5x+15-7`
`=x^(4)-2x^(3) +3x^(2)-5x+8`
when we divide `p(x) ` by x+2 then we get the remainder p(-2),
Now , `P(-2)=(-2)^(4)-2(-2)^(3)+3(-2)^(3) +3(-2)^(2)-5(-2)+8`
`=16+16+12+10+8=62`
hence ,the value of a is 5 and remainder is 62 .