# How do you divide #(4x^4 -12x^3+5x+3)/((x^2 + 4x + 4) )#?

Long divide the coefficients to find:

#(4x^4-12x^3+5x+3)/(x^2+4x+4) = 4x^2-28x+96 + (-277x-381)/(x^2+4x+4)#

I like to long divide the coefficients, not forgetting to include

The process is similar to long division of numbers.

Write the dividend

Write the first term

Write the product

Write the second term

Write the product

Write the third term

Write the product

This remainder is shorter than the divisor and there are no more terms to bring down from the dividend, so this is our final remainder and where we stop.

We find that:

#(4x^4-12x^3+5x+3)/(x^2+4x+4) = 4x^2-28x+96 + (-277x-381)/(x^2+4x+4)#

Or if you prefer:

#4x^4-12x^3+5x+3#

#= (x^2+4x+4)(4x^2-28x+96) + (-277x-381)#

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To divide (4x^4 - 12x^3 + 5x + 3) by (x^2 + 4x + 4), we can use long division.

First, we divide the highest degree term of the numerator (4x^4) by the highest degree term of the denominator (x^2). This gives us 4x^2.

Next, we multiply the entire denominator (x^2 + 4x + 4) by the quotient we just found (4x^2), and subtract the result from the numerator. This gives us (4x^4 - 12x^3 + 5x + 3) - (4x^2 * (x^2 + 4x + 4)) = -12x^3 - 11x - 13.

We then repeat the process with the new numerator (-12x^3 - 11x - 13) and the denominator (x^2 + 4x + 4).

Dividing the highest degree term of the new numerator (-12x^3) by the highest degree term of the denominator (x^2) gives us -12x.

We multiply the entire denominator (x^2 + 4x + 4) by the new quotient (-12x), and subtract the result from the new numerator. This gives us (-12x^3 - 11x - 13) - (-12x * (x^2 + 4x + 4)) = -11x - 13.

We repeat the process again with the new numerator (-11x - 13) and the denominator (x^2 + 4x + 4).

Dividing the highest degree term of the new numerator (-11x) by the highest degree term of the denominator (x^2) gives us -11.

We multiply the entire denominator (x^2 + 4x + 4) by the new quotient (-11), and subtract the result from the new numerator. This gives us (-11x - 13) - (-11 * (x^2 + 4x + 4)) = -13 - 44x - 44.

Since the degree of the new numerator (-13 - 44x - 44) is less than the degree of the denominator (x^2 + 4x + 4), we have reached the end of the long division process.

Therefore, the quotient is 4x^2 - 12x - 11, and the remainder is -13 - 44x - 44.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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