Let's solve for the product of the binomials [tex]\((2x + 5)(3x - 4)\)[/tex] using the distributive property, often referred to as the FOIL method for binomials.
Step 1: Expand the product using the distributive property
[tex]\[
(2x + 5)(3x - 4)
\][/tex]
[tex]\[
= (2x \cdot 3x) + (2x \cdot -4) + (5 \cdot 3x) + (5 \cdot -4)
\][/tex]
Step 2: Multiply each term
1. Multiply the first terms:
[tex]\[
2x \cdot 3x = 6x^2
\][/tex]
2. Multiply the outer terms:
[tex]\[
2x \cdot -4 = -8x
\][/tex]
3. Multiply the inner terms:
[tex]\[
5 \cdot 3x = 15x
\][/tex]
4. Multiply the last terms:
[tex]\[
5 \cdot -4 = -20
\][/tex]
Step 3: Combine like terms
Now, combine the terms we obtained:
[tex]\[
6x^2 - 8x + 15x - 20
\][/tex]
Combine the [tex]\(x\)[/tex] terms:
[tex]\[
-8x + 15x = 7x
\][/tex]
Step 4: Write the final expression
[tex]\[
6x^2 + 7x - 20
\][/tex]
So, the product of the binomials [tex]\((2x + 5)(3x - 4)\)[/tex] is:
[tex]\[
6x^2 + 7x - 20
\][/tex]
