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The expression in the question is \( 14m^2 + 11m - 15 \), and you are asked to factor it.

To factor a quadratic expression of the form \( ax^2 + bx + c \), we look for two binomials:

\[
(Am + B)(Cm + D)
\]

Here’s the step-by-step process:

1. Multiply the coefficient of \( m^2 \) (14) by the constant (-15), which gives us \( 14 \times -15 = -210 \).
2. Now, we need to find two numbers that multiply to give -210 and add up to the middle term coefficient, which is 11. The two numbers that fit are 21 and -10 (since \( 21 \times -10 = -210 \) and \( 21 + (-10) = 11 \)).
3. Rewrite the middle term using these two numbers: 

\[
14m^2 + 21m - 10m - 15
\]

4. Now, factor by grouping:

\[
(14m^2 + 21m) - (10m + 15)
\]

Factor out the common terms from each group:

\[
7m(2m + 3) - 5(2m + 3)
\]

5. Now, factor out the common binomial \( (2m + 3) \):

\[
(7m - 5)(2m + 3)
\]

So the factorized form of \( 14m^2 + 11m - 15 \) is \( (7m - 5)(2m + 3) \).

The correct answer is the first option: \( (7m - 5)(2m + 3) \).

Would you like more details or have any questions?

Here are 5 related questions you might find helpful:
1. How do you factor quadratic expressions when the leading coefficient is greater than 1?
2. Can all quadratic equations be factored easily?
3. What is the difference between factoring by grouping and using the quadratic formula?
4. How do you handle special cases like perfect square trinomials in factoring?
5. What are the uses of factored quadratic expressions in real-world applications?

**Tip:** When factoring quadratics, always check if the quadratic can be simplified first by dividing all terms by a common factor.

Factorize the expression: 14m^2 + 11m - 15.

Learn how to factor the quadratic expression 14m^2 + 11m - 15 using the factor by grouping method. This step-by-step solution covers key algebraic concepts suitable for students in Grades 9-11.

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