We are given two products and we need to subtract one from the other. First, let's compute each of these products step-by-step.
### Step 1: Compute the product of terms [tex]\(\left(\frac{4}{3} l^2 m n\right), \left(-5 l m n^2\right)\)[/tex], and [tex]\(\left(\frac{1}{3} l^2 m n\right)\)[/tex]
1. Multiply [tex]\(\left(\frac{4}{3} l^2 m n\right)\)[/tex] and [tex]\(\left(-5 l m n^2\right)\)[/tex]:
[tex]\[
\left(\frac{4}{3} l^2 m n\right) \cdot \left(-5 l m n^2\right) = \left(\frac{4}{3} \cdot -5\right) \cdot l^2 \cdot l \cdot m \cdot m \cdot n \cdot n^2
\][/tex]
[tex]\[
= \left(-\frac{20}{3}\right) \cdot l^3 \cdot m^2 \cdot n^3
\][/tex]
2. Now, multiply the result by [tex]\(\left(\frac{1}{3} l^2 m n\right)\)[/tex]:
[tex]\[
\left(-\frac{20}{3} l^3 m^2 n^3\right) \cdot \left(\frac{1}{3} l^2 m n\right) = \left(-\frac{20}{3} \cdot \frac{1}{3}\right) \cdot l^3 \cdot l^2 \cdot m^2 \cdot m \cdot n^3 \cdot n
\][/tex]
[tex]\[
= \left(-\frac{20}{9}\right) \cdot l^{5} \cdot m^{3} \cdot n^{4}
\][/tex]
### Step 2: Compute the product of terms [tex]\(\left(\frac{7}{3} l^3 m n^2\right)\)[/tex] and [tex]\(\left(\frac{2}{3} l^2 m^2 n^2\right)\)[/tex]
1. Multiply the terms directly:
[tex]\[
\left(\frac{7}{3} l^3 m n^2\right) \cdot \left(\frac{2}{3} l^2 m^2 n^2\right) = \left(\frac{7}{3} \cdot \frac{2}{3}\right) \cdot l^3 \cdot l^2 \cdot m \cdot m^2 \cdot n^2 \cdot n^2
\][/tex]
[tex]\[
= \left(\frac{14}{9}\right) \cdot l^5 \cdot m^3 \cdot n^4
\][/tex]
### Step 3: Subtract the second product from the first product
[tex]\[
\left(-\frac{20}{9} l^{5} m^{3} n^{4}\right) - \left(\frac{14}{9} l^{5} m^{3} n^{4}\right) = \frac{-20 - 14}{9} l^{5} m^{3} n^{4} = \frac{-34}{9} l^{5} m^{3} n^{4}
\][/tex]
Thus, the final result after subtracting the second product from the first is:
[tex]\[
\boxed{-\frac{34}{9} l^5 m^3 n^4}
\][/tex]
