To find the product of the binomials [tex]\(\left(\frac{1}{4} x - 6 \right)\)[/tex] and [tex]\(\left(\frac{4}{3} x - \frac{5}{4}\right)\)[/tex], we will use the distributive property to expand the expression.
Given:
[tex]\[ \left(\frac{1}{4} x - 6\right) \left(\frac{4}{3} x - \frac{5}{4}\right) \][/tex]
We need to multiply each term in the first binomial by each term in the second binomial:
[tex]\[
\left(\frac{1}{4} x\right) \left(\frac{4}{3} x\right) + \left(\frac{1}{4} x\right) \left(-\frac{5}{4}\right) + \left(-6 \right) \left(\frac{4}{3} x\right) + \left(-6 \right) \left(-\frac{5}{4}\right)
\][/tex]
Step-by-step, we calculate each product:
1. [tex]\(\left(\frac{1}{4} x\right) \left(\frac{4}{3} x\right)\)[/tex]:
[tex]\[
\frac{1}{4} \cdot \frac{4}{3} = \frac{1 \cdot 4}{4 \cdot 3} = \frac{4}{12} = \frac{1}{3}
\][/tex]
[tex]\[
\Rightarrow \left( \frac{1}{3} x^2 \right)
\][/tex]
2. [tex]\(\left(\frac{1}{4} x\right) \left(-\frac{5}{4}\right)\)[/tex]:
[tex]\[
\frac{1}{4} \cdot -\frac{5}{4} = \frac{1 \cdot -5}{4 \cdot 4} = -\frac{5}{16}
\][/tex]
[tex]\[
\Rightarrow \left( -\frac{5}{16} x \right)
\][/tex]
3. [tex]\(\left(-6\right) \left(\frac{4}{3} x\right)\)[/tex]:
[tex]\[
-6 \cdot \frac{4}{3} = \frac{-6 \cdot 4}{3} = \frac{-24}{3} = -8
\][/tex]
[tex]\[
\Rightarrow \left( -8 x \right)
\][/tex]
4. [tex]\(\left(-6\right) \left(-\frac{5}{4}\right)\)[/tex]:
[tex]\[
-6 \cdot -\frac{5}{4} = \frac{-6 \cdot -5}{4} = \frac{30}{4} = 7.5
\][/tex]
[tex]\[
\Rightarrow \left( 7.5 \right)
\][/tex]
Now, we combine all these terms:
[tex]\[
\left(\frac{1}{4} x - 6\right) \left(\frac{4}{3} x - \frac{5}{4}\right) = \frac{1}{3} x^2 - \left(\frac{5}{16} x + 8 x\right) + 7.5
\][/tex]
We can simplify the [tex]\( x \)[/tex] terms:
[tex]\[
-\frac{5}{16} x - 8 x = -\frac{5}{16} x - \frac{128}{16} x = -\frac{133}{16} x = -8.3125 x
\][/tex]
Therefore, the product can be expressed as:
[tex]\[
\left(\frac{1}{4} x - 6\right) \left(\frac{4}{3} x - \frac{5}{4}\right) = \frac{1}{3} x^2 - 8.3125 x + 7.5
\][/tex]
Thus, the final tritional form of the product is:
[tex]\[
\boxed{\frac{1}{3} x^2 - 8.3125 x + 7.5}
\][/tex]
