The rational expression `(8x^3 - 125)/(4x^2 +10x+ 25)` in its simplest form is :

Options

A. `2 x `
B. `5 `
C. `2x + 5 `
D. `2x - 5`

Answer

To simplify the rational expression \((8x^3 - 125)/(4x^2 + 10x + 25)\), we can follow these steps:

### Step 1: Factor the Numerator
The numerator \(8x^3 - 125\) is a difference of cubes. We can use the identity for factoring a difference of cubes:
\[ a^3 - b^3 = (a - b)(a^2 + ab + b^2) \]
Here, \(a = 2x\) and \(b = 5\) (since \(8x^3 = (2x)^3\) and \(125 = 5^3\)).

Using the identity:
\[ 8x^3 - 125 = (2x - 5)((2x)^2 + (2x)(5) + 5^2) \]
Calculating the terms:
\[ (2x)^2 = 4x^2, \quad (2x)(5) = 10x, \quad 5^2 = 25 \]
Thus, we have:
\[ 8x^3 - 125 = (2x - 5)(4x^2 + 10x + 25) \]

### Step 2: Write the Rational Expression
Now we can rewrite the rational expression:
\[ \frac{8x^3 - 125}{4x^2 + 10x + 25} = \frac{(2x - 5)(4x^2 + 10x + 25)}{4x^2 + 10x + 25} \]

### Step 3: Simplify the Expression
Since \(4x^2 + 10x + 25\) is present in both the numerator and the denominator, we can cancel it out (as long as \(4x^2 + 10x + 25 \neq 0\)):
\[ = 2x - 5 \]

### Final Answer
Thus, the simplest form of the given rational expression is:
\[ \boxed{2x - 5} \]

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