To find the product [tex]\( (3x + 4)(3x - 5) \)[/tex] using suitable identities, we can use the distributive property, also known as the FOIL method (First, Outer, Inner, Last) for multiplying two binomials. Let's go through this step-by-step:
1. First: Multiply the first terms of each binomial:
[tex]\[
3x \times 3x = 9x^2
\][/tex]
2. Outer: Multiply the outer terms of each binomial:
[tex]\[
3x \times (-5) = -15x
\][/tex]
3. Inner: Multiply the inner terms of each binomial:
[tex]\[
4 \times 3x = 12x
\][/tex]
4. Last: Multiply the last terms of each binomial:
[tex]\[
4 \times (-5) = -20
\][/tex]
5. Now, combine all the products:
[tex]\[
9x^2 + (-15x) + 12x + (-20)
\][/tex]
6. Simplify the expression by combining like terms:
[tex]\[
9x^2 - 15x + 12x - 20
\][/tex]
7. Combine the [tex]\( x \)[/tex]-terms:
[tex]\[
9x^2 - 3x - 20
\][/tex]
Therefore, the product of [tex]\( (3x + 4)(3x - 5) \)[/tex] is:
[tex]\[
9x^2 - 3x - 20
\][/tex]
