Question

select the correct answer.
rewrite the expression \\(\frac{x^3 + 10x^2 + 13x + 39}{x^2 + 2x + 1}\\) in the form \\(q(x) + \frac{r(x)}{h(x)}).
\\(\bigcirc\\) a. \\((x + 8) + \frac{ - 4x + 31}{x^2 + 2x + 1}\\)
\\(\bigcirc\\) b. \\((-4x + 31) + \frac{x + 8}{x^2 + 2x + 1}\\)
\\(\bigcirc\\) c. \\((x + 8) + \frac{ - 4x + 31}{x^3 + 10x^2 + 13x + 39}\\)
\\(\bigcirc\\) d. \\((-4x + 31) + \frac{x + 8}{x^3 + 10x^2 + 13x + 39}\\)

Explanation:

Step1: Polynomial long division setup

Divide $x^3 + 10x^2 + 13x + 30$ by $x^2 + 2x + 1$.

Step2: Divide leading terms

$\frac{x^3}{x^2} = x$, this is the first term of $q(x)$. Multiply divisor by $x$:
$x(x^2 + 2x + 1) = x^3 + 2x^2 + x$

Step3: Subtract from dividend

$(x^3 + 10x^2 + 13x + 30) - (x^3 + 2x^2 + x) = 8x^2 + 12x + 30$

Step4: Divide new leading terms

$\frac{8x^2}{x^2} = 8$, add to $q(x)$. Multiply divisor by 8:
$8(x^2 + 2x + 1) = 8x^2 + 16x + 8$

Step5: Subtract to get remainder

$(8x^2 + 12x + 30) - (8x^2 + 16x + 8) = -4x + 31$

Step6: Write final form

$q(x) = x + 8$, remainder $r(x) = -4x + 31$, divisor $b(x) = x^2 + 2x + 1$

Answer:

A. $(x + 8) + \frac{-4x + 31}{x^2 + 2x + 1}$