We need to solve the equation \( 16(x^2 - 4) + 5 = -10 \).

### Step 1: Simplify the equation
First, distribute the \( 16 \) across the expression inside the parentheses:
\[
16(x^2 - 4) + 5 = -10
\]
\[
16x^2 - 64 + 5 = -10
\]

### Step 2: Combine like terms
Now, combine the constants on the left-hand side:
\[
16x^2 - 59 = -10
\]

### Step 3: Move the constant from the right-hand side
Add \( 59 \) to both sides of the equation to isolate the term with \( x^2 \):
\[
16x^2 = 49
\]

### Step 4: Solve for \( x^2 \)
Divide both sides by 16:
\[
x^2 = \frac{49}{16}
\]

### Step 5: Take the square root of both sides
Taking the square root of both sides gives us two possible solutions for \( x \):
\[
x = \pm \frac{7}{4}
\]

Thus, the solutions are \( x = \frac{7}{4} \) and \( x = -\frac{7}{4} \).

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

Here are some related questions:
1. What are the steps to solving quadratic equations?
2. How can you check if the solutions for \( x \) are correct?
3. What are the properties of square roots when solving equations?
4. Can this equation be factored instead of simplified?
5. What is the significance of the discriminant in quadratic equations?

**Tip:** When solving for \( x^2 \), always remember that there are typically two solutions, \( + \) and \( - \), unless specified otherwise.

Learn how to solve the quadratic equation 16(x^2 − 4) + 5 = −10 with a detailed step-by-step solution. This problem involves algebraic concepts, including expanding expressions, combining like terms, and using the square root property. Suitable for students in Grades 9-10.

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