https://file.aiquickdraw.com/mathbot/file/1726815914098cljhqeat.jpg

Let's go through the factoring problems from the image one by one.

### 1. \( x^2 - 2x - 15 = 0 \)

This is a quadratic equation. We need to find two numbers whose product is \(-15\) and whose sum is \(-2\). These numbers are \(-5\) and \(3\).

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

### 2. \( x^2 + 3x - 28 = 0 \)

Again, we find two numbers whose product is \(-28\) and whose sum is \(3\). These numbers are \(7\) and \(-4\).

\[
x^2 + 3x - 28 = (x + 7)(x - 4)
\]

### 3. \( x^2 - 49 = 0 \)

This is a difference of squares, which factors as:

\[
x^2 - 49 = (x - 7)(x + 7)
\]

### 4. \( x^2 + 7x + 10 = 0 \)

We need two numbers whose product is \(10\) and whose sum is \(7\). These numbers are \(5\) and \(2\).

\[
x^2 + 7x + 10 = (x + 5)(x + 2)
\]

### 5. \( x^2 - 4x - 21 = 0 \)

We need two numbers whose product is \(-21\) and whose sum is \(-4\). These numbers are \(-7\) and \(3\).

\[
x^2 - 4x - 21 = (x - 7)(x + 3)
\]

### 6. \( x^2 - 225 = 0 \)

This is another difference of squares:

\[
x^2 - 225 = (x - 15)(x + 15)
\]

### 7. \( 9x^2y^4z^8 - 27xy^3z^4 \)

First, factor out the common terms. The greatest common factor (GCF) is \( 9xy^3z^4 \):

\[
9x^2y^4z^8 - 27xy^3z^4 = 9xy^3z^4(xz^4 - 3)
\]

### 8. \( 12x^3 - 16x^6 \)

Factor out the greatest common factor, which is \( 4x^3 \):

\[
12x^3 - 16x^6 = 4x^3(3 - 4x^3)
\]

### 9. \( x^5 + x^7 \)

Factor out the greatest common factor, which is \( x^5 \):

\[
x^5 + x^7 = x^5(1 + x^2)
\]

### 10. \( 6xy + 24x^2y \)

Factor out the greatest common factor, which is \( 6xy \):

\[
6xy + 24x^2y = 6xy(1 + 4x)
\]

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Would you like to discuss any particular steps in more detail or have any further questions?

Here are 5 related questions for further practice:

1. How do you factor a cubic equation?
2. How can you determine if a quadratic can be factored?
3. What is the general form for factoring a difference of squares?
4. How do you factor trinomials where the leading coefficient is not 1?
5. How do you handle factoring with multiple variables like in problem 7?

**Tip:** Always check for common factors first when factoring any expression.

Factor the following quadratic and polynomial equations provided in the image.

Learn how to factor quadratic equations like x^2 - 2x - 15 = 0 and polynomials like 9x^2y^4z^8 - 27xy^3z^4 with step-by-step solutions. Suitable for high school students learning algebra and polynomial factorization.

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