To factor the quadratic expression \( 3x^2 - 14x - 5 \), follow these steps:

### Step 1: Multiply the leading coefficient by the constant term.
The leading coefficient is 3, and the constant term is -5. Multiply them:
\[
3 \times (-5) = -15
\]

### Step 2: Find two numbers that multiply to -15 and add up to -14.
The two numbers are -15 and 1, since:
\[
-15 \times 1 = -15 \quad \text{and} \quad -15 + 1 = -14
\]

### Step 3: Rewrite the middle term using these two numbers.
Rewrite \(-14x\) as \(-15x + x\):
\[
3x^2 - 15x + x - 5
\]

### Step 4: Factor by grouping.
Group the terms in pairs:
\[
(3x^2 - 15x) + (x - 5)
\]
Now factor out the greatest common factor (GCF) from each group:
\[
3x(x - 5) + 1(x - 5)
\]

### Step 5: Factor out the common binomial factor.
\[
(3x + 1)(x - 5)
\]

Thus, the factored form of \( 3x^2 - 14x - 5 \) is:
\[
(3x + 1)(x - 5)
\]

Would you like more details or have any questions?

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Here are 5 related questions:
1. How do you solve quadratic equations by factoring?
2. Can all quadratic expressions be factored?
3. What is the difference between factoring and solving a quadratic?
4. How does the discriminant relate to factoring quadratics?
5. Can factoring help solve real-world problems modeled by quadratics?

**Tip**: Always check your factored form by expanding it to ensure it matches the original expression.

Learn how to factor the quadratic expression 3x^2 - 14x - 5 with a detailed, step-by-step breakdown. This problem explores algebraic techniques like factoring by grouping, finding common factors, and using the zero product property. Ideal for students in Grades 8-10.

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