Help- this literally makes no sense

9514 1404 393

  (4x -11)

Step-by-step explanation:

The volume of a rectangular prism is the product of its length, width, and height dimensions. Given any two, the third can be found by dividing the volume by the product of the given dimensions.

  V = LWH

  V/(LW) = H

If we substitute the given information, we have ...

  (4x³ +5x² -32x -33)/((x+1)(x +3)) = <missing dimension>

Suppose the missing dimension is a factor of the form (ax+b). Then you have ...

  (ax +b)(x +1)(x +3) = 4x³ +5x² -32x -33

We could do some polynomial long division, or some synthetic division, to figure this out. However, we can take a couple of shortcuts.

The product of binomial constants must be -33:

  b(1)(3) = -33

  b = -33/3 = -11

The product of x-terms must be 4x³:

  (ax)(x)(x) = 4x³

  a = 4x³/x³ = 4

The most sensible third dimension of the figure is ...

  (4x -11)

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Check

The product of the two given factors is (x+1)(x+3) = x² +4x +3. We expect that multiplying this by (4x -11) will result in 4x³ +5x² -32x -33. We developed our answer by looking at the first and last terms of the product. We can check our answer by looking at the middle two.

The x² term of the product (4x -11)(x² +4x +3) will be ...

  -11(x²) +4x(4x) = x²(-11 +16) = 5x² . . .  matches the desired product term

The x term of the product will be ...

  -11(4x) +4x(3) = x(-44 +12) = -32x  . . .  matches the desired product term