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Let's break down and solve each equation step by step to find the degree, roots, and their multiplicities.

### Problem 1:
\[
(x + 3)^5 (x + 4)^2 (x - 4)^3 = 0
\]

#### Step 1: Find the roots and their multiplicities.
- **Root** \(x = -3\) from \((x + 3)^5\), with multiplicity **5**.
- **Root** \(x = -4\) from \((x + 4)^2\), with multiplicity **2**.
- **Root** \(x = 4\) from \((x - 4)^3\), with multiplicity **3**.

#### Step 2: Find the degree of the polynomial.
The degree of the polynomial is the sum of the multiplicities of the factors:
\[
5 + 2 + 3 = 10
\]
So, the degree of the polynomial is **10**.

### Problem 2:
\[
2(x - 1)(x - 3)^2(x - 4)^4 = 0
\]

#### Step 1: Find the roots and their multiplicities.
- **Root** \(x = 1\) from \((x - 1)\), with multiplicity **1**.
- **Root** \(x = 3\) from \((x - 3)^2\), with multiplicity **2**.
- **Root** \(x = 4\) from \((x - 4)^4\), with multiplicity **4**.

#### Step 2: Find the degree of the polynomial.
The degree of the polynomial is the sum of the multiplicities of the factors:
\[
1 + 2 + 4 = 7
\]
So, the degree of the polynomial is **7**.

### Problem 3:
\[
x(x^2 - 3x - 10) = 0
\]

#### Step 1: Factor the quadratic \(x^2 - 3x - 10\).
We factor \(x^2 - 3x - 10\) as:
\[
x^2 - 3x - 10 = (x - 5)(x + 2)
\]
So, the equation becomes:
\[
x(x - 5)(x + 2) = 0
\]

#### Step 2: Find the roots and their multiplicities.
- **Root** \(x = 0

State the degree and find the roots of the equation and give its multiplicity.

Solve polynomial equations by factoring and determining the degree, roots, and multiplicity. This algebraic problem demonstrates how to find the polynomial degree by adding the multiplicities of the roots. Suitable for high school students in Grades 9-11.

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