Multiply and simplify (5x^2-6x-5)(5x^2+2x+9)

To multiply and simplify \((5x^2 - 6x - 5)(5x^2 + 2x + 9)\), we can expand each term and then combine like terms. 

### Step 1: Expand each term
We'll use the distributive property to multiply each term in the first polynomial by each term in the second polynomial:

\[
(5x^2 - 6x - 5)(5x^2 + 2x + 9) = 5x^2(5x^2 + 2x + 9) - 6x(5x^2 + 2x + 9) - 5(5x^2 + 2x + 9)
\]

### Step 2: Distribute each term

#### 1. Multiply \(5x^2\) by each term in \(5x^2 + 2x + 9\):
\[
5x^2 \cdot 5x^2 = 25x^4
\]
\[
5x^2 \cdot 2x = 10x^3
\]
\[
5x^2 \cdot 9 = 45x^2
\]

#### 2. Multiply \(-6x\) by each term in \(5x^2 + 2x + 9\):
\[
-6x \cdot 5x^2 = -30x^3
\]
\[
-6x \cdot 2x = -12x^2
\]
\[
-6x \cdot 9 = -54x
\]

#### 3. Multiply \(-5\) by each term in \(5x^2 + 2x + 9\):
\[
-5 \cdot 5x^2 = -25x^2
\]
\[
-5 \cdot 2x = -10x
\]
\[
-5 \cdot 9 = -45
\]

### Step 3: Combine like terms
Now, let's add up all terms:

\[
25x^4 + 10x^3 + 45x^2 - 30x^3 - 12x^2 - 54x - 25x^2 - 10x - 45
\]

Combine terms of the same degree:

1. \(x^4\) terms: \(25x^4\)
2. \(x^3\) terms: \(10x^3 - 30x^3 = -20x^3\)
3. \(x^2\) terms: \(45x^2 - 12x^2 - 25x^2 = 8x^2\)
4. \(x\) terms: \(-54x - 10x = -64x\)
5. Constant term: \(-45\)

### Final Answer:
\[
25x^4 - 20x^3 + 8x^2 - 64x - 45
\]

Would you like further explanation on any part of this process?

---

1. How do we apply the distributive property in polynomial multiplication?
2. What are some common mistakes to avoid when combining like terms?
3. How can polynomial multiplication be visualized in terms of area or geometry?
4. What are the applications of polynomial multiplication in algebra?
5. How do you recognize like terms in a polynomial?

**Tip:** When combining terms, organizing by degree from highest to lowest helps ensure accuracy.

Step-by-step guide on how to multiply and simplify the polynomial expression (5x^2 - 6x - 5)(5x^2 + 2x + 9). Covers key algebraic concepts such as polynomial expansion, distributive property, and term combination, suitable for Grades 10-12.

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