How do you divide #(x^4 + 5x^2 + 12x + 3)/( 6x+8)#?
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To divide (x^4 + 5x^2 + 12x + 3) by (6x + 8), you can use long division. Here are the steps:

Divide the first term of the numerator (x^4) by the first term of the denominator (6x). This gives you (1/6)x^3.

Multiply the entire denominator (6x + 8) by (1/6)x^3, and subtract the result from the numerator. This gives you a new numerator: (2/3)x^3 + 5x^2 + 12x + 3.

Bring down the next term from the numerator, which is 5x^2.

Divide the first term of the new numerator (2/3)x^3 by the first term of the denominator (6x). This gives you (1/9)x^2.

Multiply the entire denominator (6x + 8) by (1/9)x^2, and subtract the result from the new numerator. This gives you a new numerator: (1/3)x^2 + 12x + 3.

Bring down the next term from the numerator, which is 12x.

Divide the first term of the new numerator (1/3)x^2 by the first term of the denominator (6x). This gives you (1/18)x.

Multiply the entire denominator (6x + 8) by (1/18)x, and subtract the result from the new numerator. This gives you a new numerator: (1/3)x + 3.

Bring down the next term from the numerator, which is 3.

Divide the first term of the new numerator (1/3)x by the first term of the denominator (6x). This gives you (1/18).

Multiply the entire denominator (6x + 8) by (1/18), and subtract the result from the new numerator. This gives you a new numerator: 3.

Since the new numerator (3) is a constant term, you have reached the end of the division.
Therefore, the result of dividing (x^4 + 5x^2 + 12x + 3) by (6x + 8) is (1/6)x^3  (1/9)x^2 + (1/18)x + (3/(6x + 8)).
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