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The given quadratic expression is \( 4x^2 - 3x - 10 \). We want to factor it in the form \( (4x + A)(x + B) \).

### Step 1: Multiply the leading coefficient and the constant term
Multiply the coefficient of \( x^2 \) (which is 4) by the constant term (which is -10):

\[
4 \times (-10) = -40
\]

### Step 2: Find two numbers that multiply to -40 and add up to -3
We need two numbers that multiply to -40 and add up to -3 (the coefficient of \( x \)). These numbers are -8 and 5 because:

\[
-8 \times 5 = -40 \quad \text{and} \quad -8 + 5 = -3
\]

### Step 3: Split the middle term using these numbers
Rewrite the middle term -3x as -8x + 5x:

\[
4x^2 - 8x + 5x - 10
\]

### Step 4: Group and factor by grouping
Group the terms into two pairs:

\[
(4x^2 - 8x) + (5x - 10)
\]

Factor out the greatest common factor from each group:

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

### Step 5: Factor out the common binomial
Since \( x - 2 \) is common in both terms, factor it out:

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

Thus, the factored form of \( 4x^2 - 3x - 10 \) is:

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

### Final Answer:
\[
(4x + 5)(x - 2)
\]

Let me know if you would like a detailed explanation or have any questions. Here are some related questions:

1. How do you determine the coefficients when factoring a quadratic expression?
2. What other methods can be used to factor quadratics besides grouping?
3. How does the distributive property verify the factored form?
4. Can you factor \( 4x^2 - 3x - 10 \) using the quadratic formula?
5. How does the sign of the constant term affect the factorization?

**Tip**: Always verify the factored form by expanding it back to ensure it matches the original quadratic expression.

Factor the quadratic equation 4x^2 - 3x - 10.

Learn how to factor the quadratic equation 4x^2 - 3x - 10 using the method of factoring by grouping. This detailed step-by-step explanation covers key concepts in algebra, including factoring polynomials and the distributive property. Suitable for students in Grades 9-10.

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