Let's simplify the expression:

\[
(p^2 + 3p - 5)(3p^2 - 5p + 4) - 12p^3
\]

### Step 1: Expand the product

First, expand the product \((p^2 + 3p - 5)(3p^2 - 5p + 4)\) using the distributive property:

\[
(p^2 + 3p - 5)(3p^2 - 5p + 4) = p^2(3p^2 - 5p + 4) + 3p(3p^2 - 5p + 4) - 5(3p^2 - 5p + 4)
\]

Now, expand each term:

1. \( p^2(3p^2 - 5p + 4) = 3p^4 - 5p^3 + 4p^2 \)
2. \( 3p(3p^2 - 5p + 4) = 9p^3 - 15p^2 + 12p \)
3. \( -5(3p^2 - 5p + 4) = -15p^2 + 25p - 20 \)

### Step 2: Combine all terms

Combine all terms from the expansion:

\[
3p^4 - 5p^3 + 4p^2 + 9p^3 - 15p^2 + 12p - 15p^2 + 25p - 20
\]

Simplify by combining like terms:

\[
3p^4 + (-5p^3 + 9p^3) + (4p^2 - 15p^2 - 15p^2) + (12p + 25p) - 20
\]

\[
= 3p^4 + 4p^3 - 26p^2 + 37p - 20
\]

### Step 3: Subtract \(12p^3\)

Now subtract \(12p^3\):

\[
3p^4 + 4p^3 - 26p^2 + 37p - 20 - 12p^3
\]

Simplify:

\[
= 3p^4 + (4p^3 - 12p^3) - 26p^2 + 37p - 20
\]

\[
= 3p^4 - 8p^3 - 26p^2 + 37p - 20
\]

### Final Simplified Expression:

\[
\boxed{3p^4 - 8p^3 - 26p^2 + 37p - 20}
\]

Would you like further details or have any questions?

### Related Questions:
1. How do you expand products of polynomials?
2. What are the common methods for simplifying polynomial expressions?
3. How do you identify and combine like terms in polynomials?
4. What strategies can help in managing complex polynomial expansions?
5. How do you handle subtraction of polynomials effectively?

### Tip:
When expanding polynomials, carefully distribute each term to avoid missing any products, and systematically combine like terms to simplify accurately.

Learn how to simplify the polynomial expression (p^2 + 3p - 5)(3p^2 - 5p + 4) - 12p^3 through detailed expansion and combination of like terms. This step-by-step guide covers polynomial expansion using the distributive property, making it suitable for students in Grades 9-11.

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