How do you simplify $4b\left( { - 5b - 3} \right) - 2\left( {{b^2} - 7b - 4} \right)?$

Hint: In this question we are going to simplify the given equation.
First we are going to expand the given equation by applying the distributive property of multiplication.
Next simplify and combine like terms, we get the required result.
We can also find the values of $b$ from the given equation.

Formula used: The distributive property of multiplication states that
$a\left( {b + c} \right) = ab + ac$

Complete step-by-step solution:
In this question, we are going to simplify the given equation by using the distributive property of multiplication.
The given equation is
$4b\left( { - 5b - 3} \right) - 2\left( {{b^2} - 7b - 4} \right)$
First we are going to apply the distributive property of multiplication to the above equation.
$ \Rightarrow 4b\left( { - 5b} \right) + 4b\left( { - 3} \right) - 2\left( {{b^2}} \right) - 2\left( { - 7b} \right) - 2\left( { - 4} \right)$
Multiply the terms inside we get,
$ \Rightarrow - 20{b^2} - 12b - 2{b^2} + 14b + 8$
Simplify the above terms by adding the terms
Combine the terms that have the same variable and then add the terms have the same variable
$ \Rightarrow - 20{b^2} - 2{b^2} - 12b + 14b + 8$
On simplify we get,
$ \Rightarrow - 22{b^2} + 2b + 8$
Next we are going to find the value of $b$ from the above equation.
Then, the equation is of the quadratic form $p{x^2} + qx + r = 0$
Multiply the above equations by minus us get,
$ \Rightarrow 22{b^2} - 2b - 8 = 0$
Compare the above equation to the original quadratic equation and write the values
Here $p = 22,\,q = - 2,\,r = - 8$
We can find the roots using the quadratic formula.
Substitute those values in the quadratic formula, $b = \dfrac{{ - q \pm \sqrt {{{\left( q \right)}^2} - 4pr} }}{{2p}}$
$ \Rightarrow b = \dfrac{{ - \left( { - 2} \right) \pm \sqrt {{{\left( { - 2} \right)}^2} - 4\left( {22} \right)\left( { - 8} \right)} }}{{2\left( {22} \right)}}$
On squaring we get,
$ \Rightarrow b = \dfrac{{2 \pm \sqrt {4 + 704} }}{{44}}$
Let us add the term,
$ \Rightarrow b = \dfrac{{2 \pm \sqrt {708} }}{{44}}$
On taking squaring the term,
$ \Rightarrow b = \dfrac{{2 \pm 26.61}}{{44}}$
Let us splitting the term,
$ \Rightarrow b = \dfrac{{2 + 26.61}}{{44}},b = \dfrac{{2 - 26.61}}{{44}}$
On simplify the term and we get,
$ \Rightarrow b = \dfrac{{28.61}}{{44}},b = \dfrac{{24.61}}{{44}}$
Let us divide the term,
$ \Rightarrow b = 0.6502\,,\,0.5593$

Thus the simplified form of the equation $4b\left( { - 5b - 3} \right) - 2\left( {{b^2} - 7b - 4} \right)$ is $ - 22{b^2} + 2b + 8$.

Note: There are various methods to solve the quadratic equation. Some of them are as follows: factoring, using the square roots, completing the square and the quadratic formula.
The discriminant can be positive, zero, or negative, and this determines how many solutions are there to the given quadratic equation. The discriminant determines the nature of the roots of a quadratic equation. The word nature refers to the types of numbers the roots can be namely real, rational, irrational or imaginary.