The polynominal P is defined as $P ( x ) = x^3+ 4x^2 + 5x + 3$ The polynomial $P$ is defined as $P ( x ) = x^3+ 4x^2 + 5x + 3$

Express $P ( x )$ in the form $P ( x ) = ( x + 2 ) ( x^2+ ax + b ) + c$, where $a$, $b$ and $c$ are whole numbers.

any one tell me the idea of solution ♥ ♥ ♥
Hint: open up parentheses on the right hands side and compare coefficients:

$$(x+2)(x^2+ax+b)+c=x^3+(a+2)x^2+(b+2a)x+(2b+c)$$

Thus, for example, $\,a+2=4\;,\;\;b+2a=5\,$ and etc.
It is $(x+2)(x^2+2x+1)+1$. 

$x^3+4x^2+5x+3$

$=x^2(x+2)+2x^2+5x+3$

$=x^2(x+2)+2x(x+2)+x+3$

$=x^2(x+2)+2x(x+2)+(x+2)+1$

$=(x+2)(x^2+2x+1)+1$