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What is the quotient of (5x4−10x3−13x2+12x+30)÷(x2−4x+4) ?(1 point) Responses 5x2+10x+7 5 x squared plus 10 x plus 7 5x4−10x3−14x2+16x+26 5 x superscript 4 baseline minus 10 x cubed minus 14 x squared plus 16 x plus 26 5x3+10x2+7x+2 5 x cubed plus 10 x squared plus 7 x plus 2 5x2+10x+7+2x2−4x+4

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To find the quotient of \((5x^4 - 10x^3 - 13x^2 + 12x + 30) \div (x^2 - 4x + 4)\), we can use polynomial long division.

  1. Set up the division by writing \(5x^4 - 10x^3 - 13x^2 + 12x + 30\) under the long division and \(x^2 - 4x + 4\) outside.

  2. Divide the leading terms:

    • Divide \(5x^4\) by \(x^2\) to get \(5x^2\).

  3. Multiply \(5x^2\) by \(x^2 - 4x + 4\): \[ 5x^2(x^2 - 4x + 4) = 5x^4 - 20x^3 + 20x^2 \]

  4. Subtract this from the original polynomial: \[ (5x^4 - 10x^3 - 13x^2 + 12x + 30) - (5x^4 - 20x^3 + 20x^2) \] This simplifies to: \[ (0 + 10x^3 - 33x^2 + 12x + 30) \]

  5. Repeat the process. Divide \(10x^3\) by \(x^2\) to get \(10x\).

  6. Multiply \(10x\) by \(x^2 - 4x + 4\): \[ 10x(x^2 - 4x + 4) = 10x^3 - 40x^2 + 40x \]

  7. Subtract this: \[ (10x^3 - 33x^2 + 12x + 30) - (10x^3 - 40x^2 + 40x) \] This simplifies to: \[ (0 + 7x^2 - 28x + 30) \]

  8. Repeat again. Divide \(7x^2\) by \(x^2\) to get \(7\).

  9. Multiply \(7\) by \(x^2 - 4x + 4\): \[ 7(x^2 - 4x + 4) = 7x^2 - 28x + 28 \]

  10. Subtract: \[ (7x^2 - 28x + 30) - (7x^2 - 28x + 28) \] This simplifies to: \[ 0 + 0 + 2 = 2 \]

Putting this all together yields: \[ \frac{5x^4 - 10x^3 - 13x^2 + 12x + 30}{x^2 - 4x + 4} = 5x^2 + 10x + 7 + \frac{2}{x^2 - 4x + 4} \]

From the given responses, the quotient we computed, without the remainder, matches: 5x² + 10x + 7.

Thus, the final answer is: 5x² + 10x + 7.