To solve the problem of multiplying the binomials [tex]\( (3x + 4)(5x - 2) \)[/tex], we will use the distributive property (also known as the FOIL method in binomial multiplication). Here's a detailed, step-by-step solution:
1. Apply the Distributive Property:
The distributive property states that [tex]\( (a + b)(c + d) = ac + ad + bc + bd \)[/tex].
Substituting from the given binomials, we have:
[tex]\[
(3x + 4)(5x - 2)
\][/tex]
Let's distribute each term in the first binomial to every term in the second binomial:
2. First Term's Products:
Multiply the first term of the first binomial by each term in the second:
[tex]\[
3x \cdot 5x + 3x \cdot (-2)
\][/tex]
This gives us:
[tex]\[
15x^2 + (-6x)
\][/tex]
3. Second Term's Products:
Multiply the second term of the first binomial by each term in the second:
[tex]\[
4 \cdot 5x + 4 \cdot (-2)
\][/tex]
This gives us:
[tex]\[
20x + (-8)
\][/tex]
4. Combine Like Terms:
Now add all the products together:
[tex]\[
15x^2 + (-6x) + 20x + (-8)
\][/tex]
Combine the [tex]\( x \)[/tex]-terms:
[tex]\[
15x^2 + 14x - 8
\][/tex]
Therefore, the product of the binomials [tex]\( (3x + 4)(5x - 2) \)[/tex] is:
[tex]\[
15x^2 + 14x - 8
\][/tex]
The correct answer is:
C. [tex]\( 15x^2 + 14x - 8 \)[/tex]
