Question

express the product of \\(\left(6x + \frac{1}{2}\
ight)\\) and \\(\left(5x - \frac{5}{4}\
ight)\\) as a trinomial in simplest form.

Explanation:

Step1: Apply the distributive property (FOIL method)

We need to multiply each term in the first binomial \((6x + \frac{1}{2})\) by each term in the second binomial \((5x - \frac{5}{4})\).
First terms: \(6x \times 5x = 30x^2\)
Outer terms: \(6x \times (-\frac{5}{4}) = -\frac{30}{4}x = -\frac{15}{2}x\)
Inner terms: \(\frac{1}{2} \times 5x = \frac{5}{2}x\)
Last terms: \(\frac{1}{2} \times (-\frac{5}{4}) = -\frac{5}{8}\)

Step2: Combine like terms

Now, we combine the like terms (the \(x\)-terms):
\(-\frac{15}{2}x + \frac{5}{2}x = (-\frac{15}{2} + \frac{5}{2})x = -\frac{10}{2}x = -5x\)

Step3: Write the trinomial

Putting it all together, we have:
\(30x^2 - 5x - \frac{5}{8}\)

Answer:

\(30x^2 - 5x - \frac{5}{8}\)