https://r2.erweima.ai/mathbot/image/17259392051234esr46m6.jpg

The expression in the image you uploaded appears to be:

\[
3(2x - 1)^2(2)(x + 3)^{1/2} + (2x - 1)^3 \left(\frac{4}{2}\right) (x + 3)^{-1/2}
\]

### Let's simplify the expression step by step.

1. **First term**:
   \[
   3(2x - 1)^2(2)(x + 3)^{1/2} = 6(2x - 1)^2(x + 3)^{1/2}
   \]

2. **Second term**:
   \[
   (2x - 1)^3 \left(\frac{4}{2}\right) (x + 3)^{-1/2} = 2(2x - 1)^3(x + 3)^{-1/2}
   \]

### So, the expression becomes:
\[
6(2x - 1)^2(x + 3)^{1/2} + 2(2x - 1)^3(x + 3)^{-1/2}
\]

This expression is now in its simplified form.

If you're looking for further factorization, we can try factoring common terms between the two components.

#### Factor common terms:
- Both terms have a factor of \((2x - 1)^2\).
- We can factor out the \((2x - 1)^2\) term:
  \[
  (2x - 1)^2 \left[6(x + 3)^{1/2} + 2(2x - 1)(x + 3)^{-1/2}\right]
  \]

This is the fully factored form of the expression.

Would you like any further details or explanation about this factorization process?

---

### Related questions to expand your understanding:
1. How can this factorization be useful when solving equations?
2. Can this expression be integrated or differentiated in its current form?
3. What is the behavior of the expression as \( x \to \infty \)?
4. How would you handle simplifying expressions with more complex exponents?
5. What happens to this expression if \( x + 3 \) approaches zero?

#### Tip:
When factoring expressions, always check for common terms and powers, as this can make further operations like solving or integrating much easier.

Explore the step-by-step process of factoring a complex polynomial expression with exponents and radicals. This solution provides detailed explanations of algebraic concepts, including the distributive property and laws of exponents, suitable for high school students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation