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What is the product of the following two polynomials?

(2x3+3x2)(4x4−5x3−6x2)

(1 point)
Responses

8x7+2x6−27x5−18x4
8 x superscript 7 baseline plus 2 x superscript 6 baseline minus 27 x superscript 5 baseline minus 18 x superscript 4 baseline

8x7+22x6−27x5+18x4
8 x superscript 7 baseline plus 22 x superscript 6 baseline minus 27 x superscript 5 baseline plus 18 x superscript 4 baseline

8x7+2x6−3x5−18x4
8 x superscript 7 baseline plus 2 x superscript 6 baseline minus 3 x superscript 5 baseline minus 18 x superscript 4 baseline

8x7−22x6−27x5−18x4
8 x superscript 7 baseline minus 22 x superscript 6 baseline minus 27 x superscript 5 baseline minus 18 x superscript 4 baseline

1 answer

To find the product of the two polynomials \((2x^3 + 3x^2)(4x^4 - 5x^3 - 6x^2)\), we will distribute each term in the first polynomial to each term in the second polynomial.

  1. Distributing \(2x^3\):

    • \(2x^3 \cdot 4x^4 = 8x^7\)
    • \(2x^3 \cdot (-5x^3) = -10x^6\)
    • \(2x^3 \cdot (-6x^2) = -12x^5\)

  2. Distributing \(3x^2\):

    • \(3x^2 \cdot 4x^4 = 12x^6\)
    • \(3x^2 \cdot (-5x^3) = -15x^5\)
    • \(3x^2 \cdot (-6x^2) = -18x^4\)

Next, we combine all of these results: \[ 8x^7 + (-10x^6 + 12x^6) + (-12x^5 - 15x^5) - 18x^4 \]

This simplifies to:

  • Combine the \(x^6\) terms: \(-10x^6 + 12x^6 = 2x^6\)
  • Combine the \(x^5\) terms: \(-12x^5 - 15x^5 = -27x^5\)

Thus, we have: \[ 8x^7 + 2x^6 - 27x^5 - 18x^4 \]

Therefore, the product of the two polynomials is: \[ \boxed{8x^7 + 2x^6 - 27x^5 - 18x^4} \]