Find the product of the following:
$\left(5 m+3 m n+6 n^2\right)\left(-3 m n+4 n^2\right)$
\((5m)(-3mn) + (5m)(4n^2) + (3mn)(-3mn) + (3mn)(4n^2) + (6n^2)(-3mn) + (6n^2)(4n^2)\).
The goal is to expand the product of a trinomial and a binomial: \((5m + 3mn + 6n^2)(-3mn + 4n^2)\). To find the result, we use the distributive property, which allows us to multiply each term in the first expression by each term in the second expression and then sum the resulting products.
We multiply each term of the trinomial \((5m + 3mn + 6n^2)\) by each term of the binomial \((-3mn + 4n^2)\) individually:
\(5m \times (-3mn) = -15m^2n\)
\(5m \times (4n^2) = 20mn^2\)
\(3mn \times (-3mn) = -9m^2n^2\)
\(3mn \times (4n^2) = 12mn^3\)
\(6n^2 \times (-3mn) = -18mn^3\)
\(6n^2 \times (4n^2) = 24n^4\)
Now, we collect all the individual products calculated in the previous step:
\(-15m^2n + 20mn^2 - 9m^2n^2 + 12mn^3 - 18mn^3 + 24n^4\)
Next, we identify any like terms to simplify the expression. Observing the terms:
Putting it all together, the expanded polynomial is \(-15m^2n + 20mn^2 - 9m^2n^2 - 6mn^3 + 24n^4\).
The final result of the multiplication \((5m + 3mn + 6n^2)(-3mn + 4n^2)\) is \(-15m^2n + 20mn^2 - 9m^2n^2 - 6mn^3 + 24n^4\). \(\boxed{-15m^2n + 20mn^2 - 9m^2n^2 - 6mn^3 + 24n^4}\).
Expanding algebraic expressions is fundamental in engineering and physics, where you often need to simplify complex equations describing physical systems. For example, when calculating the energy distribution or stress analysis on a structural beam, you might encounter polynomial expressions representing force fields that need to be expanded and rearranged to solve for unknown variables or identify peak values. Mastering this skill ensures you can manipulate these models accurately to predict outcomes in real-world engineering projects.