https://r2.erweima.ai/mathbot/image/17259210581895x3utiba.jpg

Let's tackle the three problems from the image:

### Problem 4: 
\[
8(p - 9) + (9 - p)^2
\]

We will first expand and simplify this expression.

### Problem 5:
\[
3a^2 - 10b - 6ab + 5a
\]

We will factor this expression as much as possible.

### Problem 6:
\[
-12 + 36x^3 - 9x^2 + 48x
\]

We will factor out common terms in this expression.

I'll begin by solving each problem step-by-step.

---

**Problem 4:**
1. Expand the square: 
   \[
   (9 - p)^2 = (9 - p)(9 - p) = 81 - 18p + p^2
   \]
2. Now substitute this into the original equation:
   \[
   8(p - 9) + 81 - 18p + p^2
   \]
3. Expand \(8(p - 9)\):
   \[
   8p - 72
   \]
4. Combine like terms:
   \[
   8p - 72 + 81 - 18p + p^2
   \]
   Simplifies to:
   \[
   p^2 - 10p + 9
   \]

So, the simplified expression for Problem 4 is:
\[
p^2 - 10p + 9
\]

---

**Problem 5:**
The expression is:
\[
3a^2 - 10b - 6ab + 5a
\]

Group the terms to factor:
\[
(3a^2 + 5a) + (-10b - 6ab)
\]
Factor each group:
\[
a(3a + 5) - 2b(5 + 3a)
\]

Notice the common factor \((3a + 5)\), so factor it out:
\[
(3a + 5)(a - 2b)
\]

The factored form of Problem 5 is:
\[
(3a + 5)(a - 2b)
\]

---

**Problem 6:**
The expression is:
\[
-12 + 36x^3 - 9x^2 + 48x
\]

Factor out the greatest common factor, which is 3:
\[
3(-4 + 12x^3 - 3x^2 + 16x)
\]

Now, let's further factor the quadratic expression inside. We have:
\[
12x^3 - 3x^2 + 16x - 4
\]

Group terms:
\[
(12x^3 - 3x^2) + (16x - 4)
\]

Factor each group:
\[
3x^2(4x - 1) + 4(4x - 1)
\]

Now factor out \((4x - 1)\):
\[
(4x - 1)(3x^2 + 4)
\]

So, the fully factored form of Problem 6 is:
\[
3(4x - 1)(3x^2 + 4)
\]

---

Would you like a more detailed explanation for any of these steps? Here are five related problems to expand your practice:

1. Simplify the expression: \(6(p - 2) + (2 - p)^2\).
2. Factor \(4a^2 - 7ab - 2b^2\).
3. Expand and simplify \((x + 3)(x^2 - 2x + 4)\).
4. Factor the expression \(5x^3 + 10x^2 - 25x\).
5. Solve \(x^2 + 5x + 6 = 0\) by factoring.

**Tip:** Always check for common factors before attempting to factor more complex expressions.

Solve the expressions: 8(p - 9) + (9 - p)^2, 3a^2 - 10b - 6ab + 5a, and -12 + 36x^3 - 9x^2 + 48x.

This guide provides step-by-step solutions for factoring and simplifying polynomial expressions, including the problems 8(p - 9) + (9 - p)^2, 3a^2 - 10b - 6ab + 5a, and -12 + 36x^3 - 9x^2 + 48x. Suitable for students in Grades 9-10.

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