The problem states that $(2x+3)$ is one factor of the quadratic expression $(2x^2 - 7x - 15)$. We are asked to find the other factor.

1. Problem Description

The problem states that $(2x+3)$ is one factor of the quadratic expression $(2x^2 - 7x - 15)$. We are asked to find the other factor.

2. Solution Steps

Let the other factor be $(ax+b)$. Then we have:

$$(2x+3)(ax+b) = 2x^2 - 7x - 15$$

Expanding the left side gives:

$$2ax^2 + 2bx + 3ax + 3b = 2ax^2 + (2b+3a)x + 3b = 2x^2 - 7x - 15$$

Comparing the coefficients of the corresponding terms, we have the following system of equations:

$$2a = 2$$

$$2b + 3a = -7$$

$$3b = -15$$

From the first equation, we get:

$$a = \frac{2}{2} = 1$$

From the third equation, we get:

$$b = \frac{-15}{3} = -5$$

Substituting $a=1$ and $b=-5$ into the second equation gives:

$$2(-5) + 3(1) = -10 + 3 = -7$$

Thus, the other factor is $(x-5)$.

We can verify this by multiplying $(2x+3)$ and $(x-5)$:

$$(2x+3)(x-5) = 2x^2 - 10x + 3x - 15 = 2x^2 - 7x - 15$$

3. Final Answer

The other factor is $x-5$.