Question

the polynomial function ( q(x) = 2x^4 - 9x^3 - 12x^2 + 29x + 30 ) has known factors ( (x - 2) ) and ( (x + 1) ). which expression represents ( q(x) ) as the product of linear factors?
a. ( (2x - 5)(x + 3)(x - 2)(x + 1) )
b. ( (2x + 3)(x - 5)(x - 2)(x + 1) )
c. ( (2x + 15)(x - 1)(x - 2)(x + 1) )
d. ( (2x - 15)(x + 1)(x - 2)(x + 1) )

Explanation:

Step1: Use polynomial division or factor theorem

We know that \(q(x)\) has factors \((x - 2)\) and \((x + 1)\). Let's first perform polynomial division or use synthetic division to factor out these known factors. But maybe a better way is to expand each option and see which one matches \(q(x)=2x^{4}-9x^{3}-12x^{2}+29x + 30\).

Step2: Expand Option B: \((2x + 3)(x - 5)(x - 2)(x + 1)\)

First, multiply \((2x + 3)(x - 5)\):
\[

$$\begin{align*} (2x+3)(x - 5)&=2x\times x+2x\times(-5)+3\times x + 3\times(-5)\\ &=2x^{2}-10x + 3x-15\\ &=2x^{2}-7x-15 \end{align*}$$

\]
Then multiply \((x - 2)(x + 1)\):
\[

$$\begin{align*} (x - 2)(x + 1)&=x\times x+x\times1-2\times x-2\times1\\ &=x^{2}+x-2x - 2\\ &=x^{2}-x - 2 \end{align*}$$

\]
Now multiply the two quadratics: \((2x^{2}-7x - 15)(x^{2}-x - 2)\)
\[

$$\begin{align*} &2x^{2}(x^{2}-x - 2)-7x(x^{2}-x - 2)-15(x^{2}-x - 2)\\ &=2x^{4}-2x^{3}-4x^{2}-7x^{3}+7x^{2}+14x-15x^{2}+15x + 30\\ &=2x^{4}+(-2x^{3}-7x^{3})+(-4x^{2}+7x^{2}-15x^{2})+(14x + 15x)+30\\ &=2x^{4}-9x^{3}-12x^{2}+29x + 30 \end{align*}$$

\]
Which matches the given polynomial \(q(x)\). Let's just check one more option to be sure.

Step3: Expand Option A: \((2x - 5)(x + 3)(x - 2)(x + 1)\)

Multiply \((2x - 5)(x + 3)\):
\[

$$\begin{align*} (2x-5)(x + 3)&=2x\times x+2x\times3-5\times x-5\times3\\ &=2x^{2}+6x-5x - 15\\ &=2x^{2}+x-15 \end{align*}$$

\]
Multiply \((x - 2)(x + 1)=x^{2}-x - 2\) (from before)
Now multiply \((2x^{2}+x - 15)(x^{2}-x - 2)\)
\[

$$\begin{align*} &2x^{2}(x^{2}-x - 2)+x(x^{2}-x - 2)-15(x^{2}-x - 2)\\ &=2x^{4}-2x^{3}-4x^{2}+x^{3}-x^{2}-2x-15x^{2}+15x + 30\\ &=2x^{4}+(-2x^{3}+x^{3})+(-4x^{2}-x^{2}-15x^{2})+(-2x + 15x)+30\\ &=2x^{4}-x^{3}-20x^{2}+13x + 30 \end{align*}$$

\]
This does not match \(q(x)\), so Option A is wrong.

We can see that Option B expands to the given polynomial.

Answer:

B. \((2x + 3)(x - 5)(x - 2)(x + 1)\)