Solve 5x(x+2)(x+5) = 5x³: Comparing Factored and Expanded Forms

Let's examine the given equation:

$5x(x+2)(x+5)=5x^3$

We'll start by opening the second and third pairs of parentheses from the left (marked with an underline below) which are in the left side using the extended distribution law, the result will be placed in new parentheses (since the entire expression is multiplied by the expression that these parentheses multiply) then we'll simplify the expression in the resulting parentheses:

$5x\underline{(x+2)(x+5)}=5x^3 \\ \downarrow\\ 5x\underline{\textcolor{blue}{(}x^2+2x+5x+10\textcolor{blue}{)}}=5x^3\\ 5x\underline{\textcolor{blue}{(}x^2+7x+10\textcolor{blue}{)}}=5x^3\\$We'll continue and use the extended distribution law again and open the parentheses on the left side, then we'll move terms and combine like terms:

$5x(x^2+7x+10)=5x^3\\ \downarrow\\ 5x^3+35x^2+50x=5x^3\\ 35x^2+50x=0$

Note that we got a quadratic equation, which can be solved by factoring - by finding a common factor,

We'll continue and factor out the greatest common factor of the numbers and variables, which is the expression: $5x$:

$35x^2+50x=0 \\ \downarrow\\ 5x(7x+10)=0$

Now let's remember that the product of expressions equals 0 only if at least one of the expressions equals zero, therefore from this equation we get two simpler equations:

$5x=0\hspace{6pt}\text{/}:5\\ \boxed{x=0}$

or:

$7x+10=0\\ 7x=-10\hspace{6pt}\text{/}:7\\ \boxed{x=-\frac{7}{10}}$

Let's summarize the equation solving steps:

$5x(x+2)(x+5)=5x^3 \\ \downarrow\\ 5x\textcolor{blue}{(}x^2+7x+10\textcolor{blue}{)}=5x^3\\ \downarrow\\ 5x^3+35x^2+50x=5x^3\\ 35x^2+50x=0 \\ \downarrow\\ 5x(7x+10)=0\\ \downarrow\\ 5x=0\rightarrow\boxed{x=0}\\ 7x+10=0\rightarrow\boxed{x=-\frac{7}{10}}\\ \downarrow\\ \boxed{x=0,-\frac{7}{10}}$

Therefore the correct answer is answer D.