Question

simplify the polynomial expression given below.
$(2x - 1)(2x^2 + 5x + 3) + (3x + 6)$
the first step is as follows:
$2x(2x^2 + 5x + 3) - 1(2x^2 + 5x + 3) + (3x + 6)$
$\circ$ $4x^3 + 8x^2 + 4x + 3$
$\circ$ $4x^3 + 8x^2 + 14x + 9$
$\circ$ $2x^2 + 24x + 6$
done

Explanation:

Step1: Expand the products

First, expand \(2x(2x^{2}+5x + 3)\) and \(-1(2x^{2}+5x + 3)\):

• \(2x(2x^{2}+5x + 3)=2x\times2x^{2}+2x\times5x+2x\times3 = 4x^{3}+10x^{2}+6x\)
• \(-1(2x^{2}+5x + 3)=- 2x^{2}-5x - 3\)

Step2: Combine all terms

Now, combine the expanded terms with \((3x + 6)\):
\(4x^{3}+10x^{2}+6x-2x^{2}-5x - 3+3x + 6\)

Step3: Combine like terms

• For \(x^{3}\): \(4x^{3}\)
• For \(x^{2}\): \(10x^{2}-2x^{2}=8x^{2}\)
• For \(x\): \(6x-5x + 3x=4x\)
• For constants: \(-3 + 6=3\)

Putting it all together, we get \(4x^{3}+8x^{2}+4x + 3\)

Answer:

\(4x^{3}+8x^{2}+4x + 3\) (the first option)