Question

expand the expression to a polynomial in standard form:
$(2x^2 + 3x + 6)(2x^2 + x - 2)$

Explanation:

Step1: Use distributive property (FOIL for polynomials)

Multiply each term in the first polynomial by each term in the second polynomial:
\[

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

\]

Step2: Simplify each term

Calculate each product:
\[

$$\begin{align*} &4x^{4}+2x^{3}-4x^{2}+6x^{3}+3x^{2}-6x + 12x^{2}+6x-12 \end{align*}$$

\]

Step3: Combine like terms

Combine terms with the same power of \(x\):

• For \(x^{4}\): \(4x^{4}\)
• For \(x^{3}\): \(2x^{3}+6x^{3}=8x^{3}\)
• For \(x^{2}\): \(-4x^{2}+3x^{2}+12x^{2}=11x^{2}\)
• For \(x\): \(-6x + 6x = 0\)
• Constant term: \(-12\)

So the simplified polynomial is \(4x^{4}+8x^{3}+11x^{2}-12\)

Answer:

\(4x^{4}+8x^{3}+11x^{2}-12\)