Factor the expression completely. (This type of expression arises in calculus when using the "Product Rule.")
5(x2 + 5)4(2x)(x − 3)4 + (x2 + 5)5(4)(x − 3)3

We are tasked with factoring the following expression completely:

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

### Step 1: Identify Common Factors
First, we look for common factors in both terms:

- In the first term \( 5(x^2 + 5)^4(2x)(x - 3)^4 \):
  - The factors are \( 5 \), \( (x^2 + 5)^4 \), \( 2x \), and \( (x - 3)^4 \).

- In the second term \( (x^2 + 5)^5(4)(x - 3)^3 \):
  - The factors are \( (x^2 + 5)^5 \), \( 4 \), and \( (x - 3)^3 \).

The common factors between the two terms are:
- \( (x^2 + 5)^4 \) (since the lower power is \( 4 \) and it appears in both terms)
- \( (x - 3)^3 \) (since the lower power is \( 3 \) and it appears in both terms)

We can factor out these common terms:

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

### Step 2: Factor out the remaining terms
After factoring out \( (x^2 + 5)^4(x - 3)^3 \), we are left with:

\[
(x^2 + 5)^4(x - 3)^3 \left[ 5(2x)(x - 3) + (x^2 + 5)(4) \right]
\]

### Step 3: Simplify the remaining expression
Now, simplify the expression inside the brackets:

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

Start by distributing:

- \( 5(2x)(x - 3) = 10x(x - 3) = 10x^2 - 30x \)
- \( (x^2 + 5)(4) = 4x^2 + 20 \)

Now, add these two results:

\[
10x^2 - 30x + 4x^2 + 20 = 14x^2 - 30x + 20
\]

### Step 4: Final expression
The factored expression is:

\[
(x^2 + 5)^4(x - 3)^3(14x^2 - 30x + 20)
\]

### Step 5: Optional Further Factoring
The quadratic term \( 14x^2 - 30x + 20 \) can be factored if possible. First, factor out a common factor of 2:

\[
14x^2 - 30x + 20 = 2(7x^2 - 15x + 10)
\]

We check whether \( 7x^2 - 15x + 10 \) can be factored further. The discriminant is:

\[
\Delta = (-15)^2 - 4(7)(10) = 225 - 280 = -55
\]

Since the discriminant is negative, the quadratic cannot be factored further using real numbers.

### Final Factored Expression:

\[
2(x^2 + 5)^4(x - 3)^3(7x^2 - 15x + 10)
\]

This is the completely factored form of the given expression.

---

Would you like more details or have any questions?

### 5 Related Questions:
1. How do you apply the product rule in calculus?
2. What are discriminants and how do they affect factoring?
3. Can quadratic expressions with a negative discriminant be factored over the complex numbers?
4. How do you recognize common factors in polynomial expressions?
5. What are some examples of expressions that arise from applying the product rule?

### Tip:
When factoring expressions with multiple terms, always check for common factors across all terms first before simplifying.

Factor the expression completely. (This type of expression arises in calculus when using the 'Product Rule.') 5(x^2 + 5)^4(2x)(x − 3)^4 + (x^2 + 5)^5(4)(x − 3)^3

Learn how to factor the complex polynomial expression 5(x^2 + 5)^4(2x)(x − 3)^4 + (x^2 + 5)^5(4)(x − 3)^3, which arises from the Product Rule in calculus. This solution involves algebraic techniques such as identifying common factors, using the distributive property, and simplifying polynomials. Suitable for students in Grades 11-12 and Calculus I.

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