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

Solve the following equation:

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

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:

$5x\underline{(x+2)(x+5)}=5x^3 \\ \downarrow\\ 5x\underline{\textcolor{blue}{(}x^2+2x+5x+10\textcolor{blue}{)}}=5x^3\\ 5x\underline{\textcolor{blue}{(}x^2+7x+10\textcolor{blue}{)}}=5x^3\\$Continue to use the extended distribution law again and open the parentheses on the left side. Proceed to move 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 obtained a quadratic equation, which can be solved by factoring - by finding a common factor,

Continue to factor out the greatest common factor of the numbers and variables, which is the expression: $5x$:

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

Remember that the product of expressions equals 0 only if at least one of the expressions equals zero, therefore from this equation we obtain 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 various steps of the solution:

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

Therefore the correct answer is answer D.