High School

Multiply the following:



\[

\begin{array}{r}

x^2 + 4x + 2 \\

\times \quad 2x^2 + 3x - 4 \\

\hline

\square

\end{array}

\]



Choose the correct product:



A. \(2x^4 + 12x^2 - 8\)



B. \(2x^4 + 23x^2 - 10x - 8\)



C. \(3x^2 + 7x - 2\)



D. \(2x^4 + 11x^3 + 12x^2 - 10x - 8\)

Answer :

To multiply the polynomials [tex](x^2 + 4x + 2)[/tex] and [tex](2x^2 + 3x - 4)[/tex], we will use the distributive property, also known as the FOIL method for this type of expression, to ensure each term in the first polynomial is multiplied by each term in the second polynomial.

Here's how you can do it step-by-step:

  1. Multiply each term in the first polynomial by each term in the second polynomial:

    • Multiply [tex]x^2[/tex] by each term in the second polynomial:

      • [tex]x^2 \times 2x^2 = 2x^4[/tex]
      • [tex]x^2 \times 3x = 3x^3[/tex]
      • [tex]x^2 \times (-4) = -4x^2[/tex]

    • Multiply [tex]4x[/tex] by each term in the second polynomial:

      • [tex]4x \times 2x^2 = 8x^3[/tex]
      • [tex]4x \times 3x = 12x^2[/tex]
      • [tex]4x \times (-4) = -16x[/tex]

    • Multiply [tex]2[/tex] by each term in the second polynomial:

      • [tex]2 \times 2x^2 = 4x^2[/tex]
      • [tex]2 \times 3x = 6x[/tex]
      • [tex]2 \times (-4) = -8[/tex]

  2. Combine all the terms:

    [tex]2x^4 + 3x^3 - 4x^2 + 8x^3 + 12x^2 - 16x + 4x^2 + 6x - 8[/tex]

  3. Simplify by combining like terms:

    • Combine the [tex]x^3[/tex] terms: [tex]3x^3 + 8x^3 = 11x^3[/tex]
    • Combine the [tex]x^2[/tex] terms: [tex]-4x^2 + 12x^2 + 4x^2 = 12x^2[/tex]
    • Combine the [tex]x[/tex] terms: [tex]-16x + 6x = -10x[/tex]

    This gives us:

    [tex]2x^4 + 11x^3 + 12x^2 - 10x - 8[/tex]

So, the product is [tex]2x^4 + 11x^3 + 12x^2 - 10x - 8[/tex], which corresponds to option D.

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