Question

\\(\dfrac{x + 3}{x^2 - 2x + 1} + \dfrac{x}{x^2 - 3x + 2}\\)

\\(\dfrac{x}{x^2 - 4x + 4} - \dfrac{2}{x^2 - 4}\\)

Explanation:

Step1: Factor all denominators

First, factor each quadratic denominator:

• For \(x^2 - 2x + 1\), it's a perfect square trinomial: \(x^2 - 2x + 1=(x - 1)^2\)
• For \(x^2 - 3x + 2\), factor: \(x^2 - 3x + 2=(x - 1)(x - 2)\)
• For \(x^2 - 4x + 4\), perfect square: \(x^2 - 4x + 4=(x - 2)^2\)
• For \(x^2 - 4\), difference of squares: \(x^2 - 4=(x - 2)(x + 2)\)

So the expression becomes:
\[
\frac{\frac{x + 3}{(x - 1)^2}+\frac{x}{(x - 1)(x - 2)}}{\frac{x}{(x - 2)^2}-\frac{2}{(x - 2)(x + 2)}}
\]

Step2: Simplify numerator and denominator separately

Simplify numerator (\(N\)):

Find common denominator for \(\frac{x + 3}{(x - 1)^2}\) and \(\frac{x}{(x - 1)(x - 2)}\), which is \((x - 1)^2(x - 2)\)
\[

$$\begin{align*} N&=\frac{(x + 3)(x - 2)+x(x - 1)}{(x - 1)^2(x - 2)}\\ &=\frac{x^2 - 2x + 3x - 6+x^2 - x}{(x - 1)^2(x - 2)}\\ &=\frac{2x^2 - 6}{(x - 1)^2(x - 2)}\\ &=\frac{2(x^2 - 3)}{(x - 1)^2(x - 2)} \end{align*}$$

\]

Simplify denominator (\(D\)):

Find common denominator for \(\frac{x}{(x - 2)^2}\) and \(\frac{2}{(x - 2)(x + 2)}\), which is \((x - 2)^2(x + 2)\)
\[

$$\begin{align*} D&=\frac{x(x + 2)-2(x - 2)}{(x - 2)^2(x + 2)}\\ &=\frac{x^2 + 2x - 2x + 4}{(x - 2)^2(x + 2)}\\ &=\frac{x^2 + 4}{(x - 2)^2(x + 2)} \end{align*}$$

\]

Step3: Divide numerator by denominator (multiply by reciprocal)

\[
\frac{N}{D}=\frac{\frac{2(x^2 - 3)}{(x - 1)^2(x - 2)}}{\frac{x^2 + 4}{(x - 2)^2(x + 2)}}=\frac{2(x^2 - 3)}{(x - 1)^2(x - 2)}\times\frac{(x - 2)^2(x + 2)}{x^2 + 4}
\]

Step4: Simplify the expression

Cancel out \((x - 2)\) from numerator and denominator:
\[
\frac{2(x^2 - 3)(x - 2)(x + 2)}{(x - 1)^2(x^2 + 4)}
\]
Or expand \((x - 2)(x + 2)=x^2 - 4\), so:
\[
\frac{2(x^2 - 3)(x^2 - 4)}{(x - 1)^2(x^2 + 4)}
\]
Or further expand numerator: \(2(x^4 - 4x^2 - 3x^2 + 12)=2(x^4 - 7x^2 + 12)=2x^4 - 14x^2 + 24\)
So the simplified form is \(\frac{2(x^2 - 3)(x - 2)(x + 2)}{(x - 1)^2(x^2 + 4)}\) (or expanded forms)

Answer:

\(\frac{2(x^2 - 3)(x - 2)(x + 2)}{(x - 1)^2(x^2 + 4)}\) (or equivalent expanded forms)