Solve for X: (5x+1) (x+3) - 8 = 5(x+1)(x+2)

Options

A. 12
B. 13
C. 14
D. 15

Answer

To solve the equation \((5x + 1)(x + 3) - 8 = 5(x + 1)(x + 2)\), we will follow these steps:

### Step 1: Expand both sides of the equation
First, we will expand the left-hand side:

\[ (5x + 1)(x + 3) - 8 \]

Expanding \((5x + 1)(x + 3)\):

\[ = 5x \cdot x + 5x \cdot 3 + 1 \cdot x + 1 \cdot 3 \]
\[ = 5x^2 + 15x + x + 3 \]
\[ = 5x^2 + 16x + 3 \]

Now, subtract 8:

\[ 5x^2 + 16x + 3 - 8 = 5x^2 + 16x - 5 \]

Now, expanding the right-hand side:

\[ 5(x + 1)(x + 2) \]

Expanding \((x + 1)(x + 2)\):

\[ = x \cdot x + x \cdot 2 + 1 \cdot x + 1 \cdot 2 \]
\[ = x^2 + 2x + x + 2 \]
\[ = x^2 + 3x + 2 \]

Now, multiply by 5:

\[ 5(x^2 + 3x + 2) = 5x^2 + 15x + 10 \]

### Step 2: Set the expanded forms equal to each other
Now we have:

\[ 5x^2 + 16x - 5 = 5x^2 + 15x + 10 \]

### Step 3: Simplify the equation
Subtract \(5x^2\) from both sides:

\[ 16x - 5 = 15x + 10 \]

Now, subtract \(15x\) from both sides:

\[ 16x - 15x - 5 = 10 \]
\[ x - 5 = 10 \]

### Step 4: Solve for \(x\)
Add 5 to both sides:

\[ x = 10 + 5 \]
\[ x = 15 \]

### Final Answer
The solution for \(x\) is:

\[ \boxed{15} \]