Which of the following is equivalent to `(4x^(2) -8)/(2x +3)` ?

Options

A. `2x -3 + (1)/(2x +3)`
B. `2x -2`
C. `2x +3 - (1)/(2x +3)`
D. `2x +4`

Answer

To solve the expression \((4x^{2} - 8)/(2x + 3)\) and find an equivalent form, we can follow these steps:

### Step 1: Rewrite the Numerator
We start with the expression:
\[ \frac{4x^2 - 8}{2x + 3} \]
Notice that \(4x^2 - 8\) can be factored. We can factor out a common term from the numerator:
\[ 4(x^2 - 2) \]
So, we rewrite the expression as:
\[ \frac{4(x^2 - 2)}{2x + 3} \]

### Step 2: Add and Subtract Terms
Next, we want to manipulate the numerator to express it in terms of the denominator. We can add and subtract \(6x\) and \(1\) in the numerator:
\[ 4(x^2 - 2) = 4(x^2 + 3x - 3x - 2) = 4((x^2 + 3x) - (3x + 2)) \]
This can be rewritten as:
\[ 4(x^2 + 3x - 3x - 2) = 4((x^2 + 3x) - (3x + 2)) \]

### Step 3: Factor the Terms
Now, we can factor out \(2x + 3\) from the terms:
\[ = 4\left((2x)(x + \frac{3}{2}) - 3(x + \frac{3}{2})\right) \]
Factoring out \((2x + 3)\):
\[ = 4(2x + 3)(x - 3) + 1 \]

### Step 4: Rewrite the Expression
Now we can express the entire fraction:
\[ \frac{4(2x + 3)(x - 3) + 1}{2x + 3} \]
This can be split into two parts:
\[ = 4(x - 3) + \frac{1}{2x + 3} \]

### Final Expression
Thus, the final equivalent expression is:
\[ 4(x - 3) + \frac{1}{2x + 3} \]

### Summary
The equivalent expression to \(\frac{4x^2 - 8}{2x + 3}\) is:
\[ 4(x - 3) + \frac{1}{2x + 3} \]

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