Question

multiply.
$left(2x^{2} + 5x + 2
ight)left(6x^{2} - 7x - 4
ight)$
simplify your answer.

Explanation:

Step1: Apply the distributive property (FOIL for polynomials)

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

$$\begin{align*} &(2x^{2}+5x + 2)(6x^{2}-7x - 4)\\ =&2x^{2}\times(6x^{2}-7x - 4)+5x\times(6x^{2}-7x - 4)+2\times(6x^{2}-7x - 4) \end{align*}$$

\]

Step2: Multiply each term

• For \(2x^{2}\times(6x^{2}-7x - 4)\):

\[
2x^{2}\times6x^{2}=12x^{4}, \quad 2x^{2}\times(-7x)=-14x^{3}, \quad 2x^{2}\times(-4)=-8x^{2}
\]

• For \(5x\times(6x^{2}-7x - 4)\):

\[
5x\times6x^{2}=30x^{3}, \quad 5x\times(-7x)=-35x^{2}, \quad 5x\times(-4)=-20x
\]

• For \(2\times(6x^{2}-7x - 4)\):

\[
2\times6x^{2}=12x^{2}, \quad 2\times(-7x)=-14x, \quad 2\times(-4)=-8
\]

Step3: Combine like terms

• \(x^{4}\) term: \(12x^{4}\)
• \(x^{3}\) terms: \(-14x^{3}+30x^{3}=16x^{3}\)
• \(x^{2}\) terms: \(-8x^{2}-35x^{2}+12x^{2}=-31x^{2}\)
• \(x\) terms: \(-20x - 14x=-34x\)
• Constant term: \(-8\)

So the simplified polynomial is \(12x^{4}+16x^{3}-31x^{2}-34x - 8\)

Answer:

\(12x^{4}+16x^{3}-31x^{2}-34x - 8\)