How do you simplify \[5w\left( -7w+3 \right)+2w\left( -2{{w}^{2}}+19w+2 \right)\]?

Hint: In this problem, we have to simplify the given expression. We can first multiply the term 5w and 2w inside the respective brackets, we will get a separate equation. we can then add the both equations as we are given an additional sign in between the equation. We can then rearrange according to the highest power and perform addition to get a simplified form.

Complete step by step solution:
We know that the expression given to be solved is,
\[5w\left( -7w+3 \right)+2w\left( -2{{w}^{2}}+19w+2 \right)\]
We can first multiply the terms which is outside the bracket, that is 5w and 2w inside the bracket, we get
\[\Rightarrow \left( -35{{w}^{2}}+15w \right)+\left( -4{{w}^{3}}+38{{w}^{2}}+4w \right)\]
We can remove the brackets as we have addition sign to add up the two equations, we get
\[\Rightarrow -35{{w}^{2}}+15w-4{{w}^{3}}+38{{w}^{2}}+4w\]
Now we can rearrange the above step by adding the similar terms from the highest power to the constant,
\[\Rightarrow -4{{w}^{3}}-35{{w}^{2}}+38{{w}^{2}}+15w+4w\]
Now we can simplify the above step, by adding the similar terms with the variables and the constant to get a simplified form.
\[\Rightarrow -4{{w}^{3}}+3{{w}^{2}}+19w\]

Therefore, the simplified form of the given expression by adding the give two quadratic equation \[5w\left( -7w+3 \right)+2w\left( -2{{w}^{2}}+19w+2 \right)\] is \[-4{{w}^{3}}+3{{w}^{2}}+19w\].

Note: Students make mistakes while multiplying the terms outside the brackets to the terms inside, as it has the variable w, which should also be multiplied. We should perform the addition operation respective of the coefficients. We can add the terms with \[{{w}^{2}}\], and the constant term to get the final simplified form.