3𝑢2 − 10𝑢 − 8. Factor. Make sure to show all factoring out steps

To factor the quadratic expression \(3u^2 - 10u - 8\), we'll follow these steps:

### 1. Identify the coefficients
The expression is of the form \(ax^2 + bx + c\), where:
- \(a = 3\) (coefficient of \(u^2\))
- \(b = -10\) (coefficient of \(u\))
- \(c = -8\) (constant term)

### 2. Multiply \(a\) and \(c\)
Multiply the coefficient of \(u^2\) (\(a\)) by the constant term (\(c\)):

\[
3 \times -8 = -24
\]

### 3. Find two numbers that multiply to \(-24\) and add to \(-10\)
We need to find two numbers that multiply to \(-24\) and add to \(-10\). The numbers are \(-12\) and \(2\) because:

\[
-12 \times 2 = -24
\]
\[
-12 + 2 = -10
\]

### 4. Rewrite the middle term using the two numbers
Rewrite the middle term \(-10u\) using \(-12u\) and \(2u\):

\[
3u^2 - 12u + 2u - 8
\]

### 5. Factor by grouping
Group the terms in pairs and factor out the common factors:

\[
(3u^2 - 12u) + (2u - 8)
\]

Factor out the common factors from each group:

\[
3u(u - 4) + 2(u - 4)
\]

### 6. Factor out the common binomial
Since \(u - 4\) is common in both terms, factor it out:

\[
(u - 4)(3u + 2)
\]

### Final Answer
The factored form of the quadratic expression \(3u^2 - 10u - 8\) is:

\[
(u - 4)(3u + 2)
\]

Would you like more details or have any questions?

Here are 8 related questions to further explore the topic:

1. How can we verify that \((u - 4)(3u + 2)\) is the correct factorization?
2. What if the quadratic expression had a positive middle term? How would that affect the factorization process?
3. Can you apply the same factoring method to the expression \(2u^2 + 5u - 3\)?
4. How does the discriminant relate to the factorization of a quadratic expression?
5. What are the roots of the equation \(3u^2 - 10u - 8 = 0\)?
6. How would the factorization change if \(a\), \(b\), and \(c\) were all positive?
7. What other methods can be used to factor quadratic expressions?
8. How can the quadratic formula be used to find the roots of \(3u^2 - 10u - 8\)?

**Tip:** Always check your factorization by expanding the factors to ensure you get the original expression back.

Learn how to factor the quadratic expression 3u^2 - 10u - 8 with a detailed step-by-step solution. This problem covers key algebraic concepts, including quadratic equations, factoring, and polynomials. Suitable for high school students.

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