Question

the rational expression \\(\frac{x^3 - 4x^2 + 6}{x^2 + 2x}\\) can be written as \\(ax + b + \frac{cx + d}{x^2 + 2x}\\) where \\(a, b, c,\\) and \\(d\\) are constants?
what are the values of \\(c\\) and \\(d\\)?
enter your answer
\\(c = \square\\)
\\(d = \square\\)

Explanation:

Step1: Combine right-hand side

$$ ax + b + \frac{cx + d}{x^2 + 2x} = \frac{(ax + b)(x^2 + 2x) + cx + d}{x^2 + 2x} $$

Step2: Expand the numerator

$$ (ax + b)(x^2 + 2x) + cx + d = ax^3 + 2ax^2 + bx^2 + 2bx + cx + d $$

Step3: Group like terms

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

Step4: Equate coefficients with left numerator

For $x^3$: $a = 1$
For $x^2$: $2a + b = -4$
Substitute $a=1$: $2(1) + b = -4 \implies b = -6$
For $x$: $2b + c = 0$
Substitute $b=-6$: $2(-6) + c = 0 \implies c = 12$
For constant term: $d = 6$

Answer:

$c = 12$
$d = 6$