My first Original problem 1

L.H.S :

$x^4 + 2x^3 + 2x^2 + x + 1$

$=\overbrace{x^4 + x}^{\text{ even number }}+ 1 +\underbrace{ 2(x^3 + x^2 )}_{\text{even number}}$

$= 2m + 1 + 2n$

$= \underbrace{2(m+n)}_{\text{even number }} +1$

$= 2k + 1 =$ odd integer .

But the R.H.S is an even number so the number of integral solution to the given equation is $\boxed {zero}$ .