Let's simplify the expression \((3x+5)^2 - (x+1)^2\).

We can apply the difference of squares formula:

\[
a^2 - b^2 = (a - b)(a + b)
\]

Here, \(a = 3x + 5\) and \(b = x + 1\). Using the formula, we get:

\[
(3x + 5)^2 - (x + 1)^2 = \left( (3x + 5) - (x + 1) \right) \left( (3x + 5) + (x + 1) \right)
\]

### Step 1: Simplify each part

\[
(3x + 5) - (x + 1) = 3x + 5 - x - 1 = 2x + 4
\]

\[
(3x + 5) + (x + 1) = 3x + 5 + x + 1 = 4x + 6
\]

### Step 2: Multiply the two factors

Now, multiply \((2x + 4)\) and \((4x + 6)\):

\[
(2x + 4)(4x + 6) = 2x(4x + 6) + 4(4x + 6)
\]

First, distribute \(2x\):

\[
2x(4x + 6) = 8x^2 + 12x
\]

Next, distribute \(4\):

\[
4(4x + 6) = 16x + 24
\]

Now, combine like terms:

\[
8x^2 + 12x + 16x + 24 = 8x^2 + 28x + 24
\]

### Final answer:

\[
(3x + 5)^2 - (x + 1)^2 = 8x^2 + 28x + 24
\]

Would you like more details, or do you have any questions?

Here are five related questions for practice:
1. Simplify \((4x - 3)^2 - (2x + 1)^2\).
2. Expand \((2x + 5)(3x - 7)\).
3. Use the difference of squares formula to simplify \((5x + 2)^2 - (x - 4)^2\).
4. Factor the quadratic expression \(9x^2 - 16\).
5. Solve for \(x\) in the equation \( (2x + 3)^2 = 49\).

**Tip:** Always remember that the difference of squares formula is useful when simplifying expressions with squared terms!

Learn how to simplify the expression (3x+5)² - (x+1)² using the difference of squares formula. This step-by-step solution covers key algebraic concepts and is suitable for students in Grades 8-10.

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