Solve x^7 + 5x^6 = 0: Factor and Find All Solutions

The equation in the problem is:

$x^7+5x^6=0$

First, let's note that in the left side we can factor the expression using a common factor, the largest common factor for the numbers and letters in this case is $x^6$since the sixth power is the lowest power in the equation and therefore is included in both the term with the seventh power and the term with the sixth power. Any power higher than this is not included in the term with the sixth power, which is the lowest, and therefore this is the term with the highest power that can be factored out as a common factor from all terms in the expression. Let's continue and perform the factoring:

$x^7+5x^6=0 \\ \downarrow\\ x^6(x+5)=0$

Let's continue and address the fact that in the left side of the equation we got in the last step there is a multiplication of algebraic expressions and on the right side the number 0, therefore, since the only way to get 0 from multiplication is to multiply by 0, at least one of the expressions in the multiplication on the left side must equal zero,

meaning:

$x^6=0 \hspace{8pt}\text{/}\sqrt[6]{\hspace{6pt}}\\ x=\pm0\\ \boxed{x=0}$(In this case, taking the even root of the right side of the equation will indeed yield two possibilities, positive and negative, but since we're dealing with zero, we'll get only one possibility)

Or:

$x+5=0\\ \boxed{x=-5}$

Let's summarize the solution of the equation:

$x^7+5x^6=0 \\ \downarrow\\ x^6(x+5)=0\\ \downarrow\\ x^6=0 \rightarrow\boxed{ x=0}\\ x+5=0 \rightarrow \boxed{x=-5}\\ \downarrow\\ \boxed{x=0,-5}$

Therefore the correct answer is answer D.