To solve the equation 

\[
\frac{5x + 30}{x^2 + 10x + 24} + 1 = 0,
\]

we first aim to simplify and solve for \(x\).

### Step 1: Combine the Fraction

Rewrite the equation by combining the fraction and the 1 into a single fraction:

\[
\frac{5x + 30}{x^2 + 10x + 24} + 1 = \frac{5x + 30}{x^2 + 10x + 24} + \frac{x^2 + 10x + 24}{x^2 + 10x + 24} = \frac{5x + 30 + (x^2 + 10x + 24)}{x^2 + 10x + 24}.
\]

### Step 2: Simplify the Numerator

Combine like terms in the numerator:

\[
5x + 30 + x^2 + 10x + 24 = x^2 + 15x + 54.
\]

So the equation becomes:

\[
\frac{x^2 + 15x + 54}{x^2 + 10x + 24} = 0.
\]

### Step 3: Set the Numerator Equal to Zero

A fraction is equal to zero when its numerator is zero (and the denominator is not zero). Thus, we solve:

\[
x^2 + 15x + 54 = 0.
\]

### Step 4: Factor the Quadratic Equation

We factor the quadratic:

\[
x^2 + 15x + 54 = (x + 6)(x + 9) = 0.
\]

### Step 5: Find the Solutions

Set each factor equal to zero:

1. \(x + 6 = 0 \implies x = -6\)
2. \(x + 9 = 0 \implies x = -9\)

### Step 6: Check for Validity Against the Denominator

Now, ensure that these values do not make the denominator zero:

The denominator \(x^2 + 10x + 24\) factors as:

\[
x^2 + 10x + 24 = (x + 4)(x + 6).
\]

Setting each factor equal to zero:

- \(x + 4 = 0 \implies x = -4\)
- \(x + 6 = 0 \implies x = -6\)

Since \(x = -6\) makes the denominator zero, we discard it as a solution.

### Conclusion

The only valid solution is:

\[
x = -9.
\]

Do you want more details or have any questions?

Here are 5 related questions to expand on this topic:

1. How do you factor quadratics in general?
2. What happens if both solutions of the numerator make the denominator zero?
3. Can you explain why a fraction equals zero when its numerator is zero?
4. What other methods can be used to solve quadratic equations?
5. How do you determine if a rational expression is undefined?

**Tip:** Always check if potential solutions make the denominator of a fraction zero, as these are not valid in the domain of the function.

Learn how to solve the equation (5x + 30) / (x^2 + 10x + 24) + 1 = 0 step-by-step with detailed solutions. This problem involves combining fractions, solving quadratic equations, and checking validity against the denominator. Suitable for high school students in Grades 9-12.

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