How many times does (x-4) go into (x^2+4x-3)?

x+8 with a remainder of 29

(x^2 + 4x - 3) / (x - 4)

Quotient and remainder:

x^2 + 4 x - 3 = (x + 8) × (x - 4) + 29

For the explanation to the method I use, see my answer to

/question/index?qid=20200...

(x^2 + 4x - 3) / (x - 4)

step 1: x^2/x = x

step 2: x(x - 4) = x^2 - 4x

step 3: (x^2 + 4x - 3) - (x^2 - 4x) = 8x - 3 (careful with signs)

step 4: recap: (x^2+4x-3) / (x-4) = x, with a remainder of (8x - 3)

repeat the 4 steps, using the remainder

(8x - 3) / (x - 4)

step 1: 8x/x = 8

step 2: 8(x-4) = 8x - 32

step 3: (8x - 3) - (8x - 32) = +29

step 4 (recap): (x^2+4x-3) / (x-4) = x + 8, with a remainder of 29

Because the remainder is of a smaller degree than the denominator, we stop.

Final answer

(x^2+4x-3) / (x-4) = x + 8, with a remainder of 29

which can be written as

x^2 + 4x - 3 = [(x-4)(x+8)] + 29

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Depending on how one interprets the phrase

"how many times does (x-4) go into x^2 + 4x - 3"

the answer could be "(x+8) times", a phrase that makes little sense in the world of polynomials.

Be careful with this. Here is what we get from synthetic division:

x² + 4x - 3 = (x + 8)(x - 4) + 29

That does not mean (x - 4) goes into (x² + 4x - 3) (x + 8) times. Remember that x is a variable, and can be any real number, so (x + 8) is not necessarily an integer, and it may well even be negative.

Or you can find out by dividing.  Think of this question:

How many times does 5 go into 18?  The answer is 3 with a remainder and we get that by dividing.  Your question is the same question just with different numbers:

(x² + 4x - 3) / (x - 4)

Using long division:

. . . . _x_+_8_____

x - 4 ) x² + 4x - 3

. . . . . x² - 4x

. . . . ------------

. . . . . . . . 8x - 3

. . . . . . . . 8x - 32

. . . . . . . . ----------

. . . . . . . . . . . 29

So it goes in (x + 8) times with a remainder of 29.

x² + 4x - 3 = (x - 4)(x + a) + b

so, x² + 4x - 3 = x² + (a - 4)x - 4a + b

Hence, a - 4 = 4

so, a = 8

Then, -4a + b = -3

i.e. -4(8) + b = -3

so, b = 29

Therefore, x² + 4x - 3 = (x - 4)(x + 8) + 29

or, (x² + 4x - 3)/(x - 4) = (x + 8) + 29/(x - 4)

i.e. (x + 8) times with remainder of 29

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