To solve the equation \((2x + 1)(x + 5) = -3\), we first expand and simplify it. Then, we'll solve the resulting quadratic equation.

### Step 1: Expand the Equation
Expand the left side:
\[
(2x + 1)(x + 5) = 2x(x + 5) + 1(x + 5) = 2x^2 + 10x + x + 5 = 2x^2 + 11x + 5
\]
So, the equation becomes:
\[
2x^2 + 11x + 5 = -3
\]

### Step 2: Move All Terms to One Side
Add 3 to both sides to set the equation to 0:
\[
2x^2 + 11x + 5 + 3 = 0
\]
\[
2x^2 + 11x + 8 = 0
\]

### Step 3: Solve the Quadratic Equation
Now, we solve the quadratic equation \(2x^2 + 11x + 8 = 0\) using the quadratic formula:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
Here, \(a = 2\), \(b = 11\), and \(c = 8\).

Calculate the discriminant:
\[
b^2 - 4ac = 11^2 - 4(2)(8) = 121 - 64 = 57
\]

Now, substitute the values into the quadratic formula:
\[
x = \frac{-11 \pm \sqrt{57}}{4}
\]

### Final Answer:
The solutions are:
\[
x = \frac{-11 + \sqrt{57}}{4} \quad \text{and} \quad x = \frac{-11 - \sqrt{57}}{4}
\]

Would you like a detailed breakdown of the quadratic formula, or do you have any other questions?

Here are 5 related questions you might explore:
1. How do you solve a quadratic equation when the discriminant is negative?
2. Can you simplify the expression \(\sqrt{57}\) any further?
3. What does the discriminant tell us about the nature of the roots?
4. How would you solve the equation if it were \(2x^2 + 11x + 8 = 3\)?
5. What are the steps to factor a quadratic equation instead of using the quadratic formula?

**Tip:** When solving quadratic equations, always check if you can factor the equation easily before using the quadratic formula—it can save time!

Learn how to solve the equation (2x + 1)(x + 5) = -3 step-by-step with a detailed solution. This problem involves expanding and solving a quadratic equation using the quadratic formula. Suitable for high school students studying algebra.

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