To solve the expression \((3x + 5)(x - 2) + (5x - 8)(2x - 3)\), we need to expand each binomial product separately and then combine the like terms.

### Step 1: Expand \((3x + 5)(x - 2)\)

1. Multiply the first terms: \(3x \times x = 3x^2\)
2. Multiply the outer terms: \(3x \times -2 = -6x\)
3. Multiply the inner terms: \(5 \times x = 5x\)
4. Multiply the last terms: \(5 \times -2 = -10\)

So, the expansion is:

\[
3x^2 - 6x + 5x - 10
\]

Combine like terms \(-6x + 5x\):

\[
3x^2 - x - 10
\]

### Step 2: Expand \((5x - 8)(2x - 3)\)

1. Multiply the first terms: \(5x \times 2x = 10x^2\)
2. Multiply the outer terms: \(5x \times -3 = -15x\)
3. Multiply the inner terms: \(-8 \times 2x = -16x\)
4. Multiply the last terms: \(-8 \times -3 = 24\)

So, the expansion is:

\[
10x^2 - 15x - 16x + 24
\]

Combine like terms \(-15x - 16x\):

\[
10x^2 - 31x + 24
\]

### Step 3: Add the two expanded expressions

Now, combine the results from Step 1 and Step 2:

\[
(3x^2 - x - 10) + (10x^2 - 31x + 24)
\]

Combine like terms:

\[
3x^2 + 10x^2 - x - 31x - 10 + 24
\]

\[
13x^2 - 32x + 14
\]

So, the final simplified expression is:

\[
13x^2 - 32x + 14
\]

Would you like more details or have any questions?

Here are some related questions you might find interesting:

1. How do you identify like terms when adding or subtracting polynomials?
2. What is the importance of the distributive property in expanding algebraic expressions?
3. How would the expression change if the signs in the binomials were different?
4. Can you factor the final expression \(13x^2 - 32x + 14\)?
5. How can this technique be applied to solve real-world problems?

**Tip:** When expanding and combining expressions, keep track of all signs (positive and negative) to avoid errors.

Learn how to expand and simplify the expression (3x + 5)(x - 2) + (5x - 8)(2x - 3) with step-by-step instructions. This problem covers algebraic concepts such as binomial expansion and the distributive property. Suitable for students in Grades 7-9.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation