(a)To calculate: The following equation {[(3x + 5)x + 4]x + 3} x + 1 = 3x^4 + 5x^3 + 4x^2 + 3x + 1 is an identity,(b) To calculate: The lopynomial P(x) = 6x^5 - 3x^4 + 9x^3 + 6x^2 -8x + 12 without powers of x as in patr (a).

Given: A fourth-degree polynomial in x such as $3x^4+5x^3+4x^2+3x+1$ contains all of the powers of x from the first through the fourth. However, any polynomial can be written without powers of x. Evaluating a polynomial without powers of x (Horner's method) is somewhat easier than evaluating a polynomial with powers. Formula used: Expansion of a polynomial, Suppose, an equation is given in the form $a(x+2)x$ can be expanded and written as, $a(x+2)x=a{x}^2+2ax.$ This is known as expansion of an equation. Calculation: The given equation $[(3x+5)x+4]x+3x+1$ can be expanded as follow, $[(3x+5)x+4]x+3x+1=[3x^2+5x+4]x+3x+1$
$=3x^3+5x^2+4x+3x+1$
$=3x^4+5x^3+4x^2+3x+1$ Therefore, the equation $[(3x+5)x+4]x+3x+1=3x^4+5x^3+4x^2+3x+1$ isan identifity. (b) Given: A fourth-degree polynomial in x such as $3x^4+5x^3+4x^2+3x+1$ contains all of the powers of xfrom the first through the fourth. However, any polynomial can be written without powers of x. Evaluating a polynomial without powers of x (Horner's method) is somewhat easier than evaluating a polynomial with powers. Formula used: Grouping of similar terms in a polynomial, Suppose, an equation is given in the form $a{x}^2+2ax+3$ can be grouped and written as, $a{x}^2+2ax+3=ax(x+2)+3=a(x+2)x+3.$ This is known as grouping of an equation. Calculation: The polynomial, $P(x)=6x^5—3x^4+9x^3+6x^2—8x+12,$ is grouped by the similar terms and written as follow,
$P(x)=6x^5—3x^4+9x^3+6x^2—8x+12=x(6x^4-3x^3+9x^2-8x)+12$
$=x(x(6x^3-3x^2+9x+6)-8)+12$
$=x[x(x(6x^2-3x+9)+6)-8]+12$ Furhter, $x[x(x(6x^2-3x+9)+6)-8]+12=[((6x-3)x+9)x+6]x-8x+12$ Therefore, the polynomial $P(x)=6x^5—3x^4+9x^3+6x^2—8x+12$ without powers is $[((6x-3)x+9)x+6]x-8x+12$