Given the expression: 15x2 + 25x − 10Part A: What is the greatest common factor? Explain how to find it. (3 points)Part B: Factor the expression completely. Show all necessary steps. (5 points)Part C: Check your factoring from Part B by multiplying. Show all necessary steps. (2 points)

Sure, let's break down the problem step by step.

Part A: Greatest Common Factor (GCF)

To find the greatest common factor (GCF) of the expression $15x^2 + 25x - 10$, we need to identify the largest factor that is common to all the terms.

1. Identify the coefficients and constants:

   • The coefficients are 15, 25, and -10.

2. Find the GCF of the coefficients:

   • The factors of 15 are 1, 3, 5, 15.
   • The factors of 25 are 1, 5, 25.
   • The factors of -10 are 1, 2, 5, 10 (we consider the absolute value for GCF).
   • The common factor among these is 5.

3. Check for common variables:

   • The variable $x$ is present in the first two terms but not in the third term, so it is not part of the GCF.

Thus, the GCF of the expression $15x^2 + 25x - 10$ is 5.

Part B: Factor the Expression Completely

1. Factor out the GCF from the expression: $15x^2 + 25x - 10 = 5(3x^2 + 5x - 2)$

2. Factor the quadratic expression inside the parentheses:

   • We need to factor $3x^2 + 5x - 2$.
   • Look for two numbers that multiply to $3 \times -2 = -6$ and add to 5.
   • These numbers are 6 and -1.

3. Rewrite the middle term using these numbers: $3x^2 + 5x - 2 = 3x^2 + 6x - x - 2$

4. Group the terms and factor by grouping: $= (3x^2 + 6x) + (-x - 2)$ $= 3x(x + 2) - 1(x + 2)$

5. Factor out the common binomial factor: $= (3x - 1)(x + 2)$

6. Combine with the GCF: $15x^2 + 25x - 10 = 5(3x - 1)(x + 2)$

Part C: Check the Factoring

To verify the factoring, we will multiply the factors back together and check if we get the original expression.

1. Multiply the binomials: $(3x - 1)(x + 2) = 3x(x + 2) - 1(x + 2)$ $= 3x^2 + 6x - x - 2$ $= 3x^2 + 5x - 2$

2. Multiply by the GCF: $5(3x^2 + 5x - 2) = 15x^2 + 25x - 10$

Since the result matches the original expression, the factoring is correct.

Thus, the complete factorization of $15x^2 + 25x - 10$ is $5(3x - 1)(x + 2)$.