`(5x^(2)+12x+7) div (5x+7) = `________

Options

A. `x+5`
B. `x+3`
C. `x+5`
D. `x+1`

Answer

To solve the problem `(5x² + 12x + 7) ÷ (5x + 7)`, we will perform polynomial long division. Here are the steps:

### Step 1: Set up the division
Write the dividend (5x² + 12x + 7) under the long division symbol and the divisor (5x + 7) outside.

### Step 2: Divide the leading terms
Divide the leading term of the dividend (5x²) by the leading term of the divisor (5x):
\[ \frac{5x²}{5x} = x \]
This means the first term of the quotient is \( x \).

### Step 3: Multiply and subtract
Now, multiply the entire divisor (5x + 7) by the term we just found (x):
\[ x(5x + 7) = 5x² + 7x \]
Subtract this result from the original polynomial:
\[ (5x² + 12x + 7) - (5x² + 7x) = 12x - 7x + 7 = 5x + 7 \]

### Step 4: Repeat the process
Now, we have a new polynomial (5x + 7). We will repeat the process:
Divide the leading term of the new polynomial (5x) by the leading term of the divisor (5x):
\[ \frac{5x}{5x} = 1 \]
So, the next term of the quotient is \( 1 \).

### Step 5: Multiply and subtract again
Multiply the entire divisor (5x + 7) by 1:
\[ 1(5x + 7) = 5x + 7 \]
Subtract this from the new polynomial:
\[ (5x + 7) - (5x + 7) = 0 \]

### Step 6: Write the final answer
Since the remainder is 0, we can conclude that:
\[ (5x² + 12x + 7) ÷ (5x + 7) = x + 1 \]

Thus, the final answer is:
\[ \boxed{x + 1} \]