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Part (B)

Step 1: Factor 2x2+5x−32x^2 + 5x - 3.
Find two numbers that multiply to a⋅c=2⋅(−3)=−6a \cdot c = 2 \cdot (-3) = -6 and add to b=5b = 5: 66 and −1-1.

2x2+6x−x−32x^2 + 6x - x - 3

Step 2: Group terms.
2x(x+3)−1(x+3)2x(x + 3) - 1(x + 3)

Step 3: Factor out common binomial.
(2x−1)(x+3)(2x - 1)(x + 3)

(2x-1)(x+3)

Part (D)

Step 1: Simplify 2x3+x2−4x+2x2−1\dfrac{2x^3 + x^2 - 4x + 2}{x^2 - 1} using polynomial division.

Divide 2x32x^3 by x2x^2:
2x2x.

Step 2: Multiply 2x2x by x2−1x^2 - 1:
2x(x2−1)=2x3−2x2x(x^2 - 1) = 2x^3 - 2x

Step 3: Subtract:

&(2x^3 + x^2 - 4x + 2) - (2x^3 - 2x)\\ &= x^2 - 2x + 2 \end{align*}$$ Step 4: Divide $x^2$ by $x^2$: $1$. Step 5: Multiply $1$ by $x^2 - 1$: $$x^2 - 1$$ Step 6: Subtract: $$\begin{align*} &(x^2 - 2x + 2) - (x^2 - 1)\\ &= -2x + 3 \end{align*}$$ Step 7: Quotient is $2x + 1$, remainder $-2x + 3$. $$\dfrac{2x^3 + x^2 - 4x + 2}{x^2 - 1} = 2x + 1 + \dfrac{-2x + 3}{x^2 - 1}$$ **Final answer:** **2x + 1 + \dfrac{-2x + 3{x^{2} - 1}}**
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