$$-\frac{a}{5}\left(2\times\frac{a^2}{5}\right)^2+5\left(-\frac{a}{5}\right)^3\left(2\times\frac{a^2}{5}\right) -a^2\left(-\frac{a}{5}\right)\left(2\times\frac{a^2}{5}\right)=0$$
I get
$$\frac{-4a^5}{25}-\frac{2a^5}{125}+\frac{2a^5}{25}$$
Where I multiply the first and third fraction by 5 in the numerator and denominator to get 125 in the denominator
For some reason, I get the wrong answer of
$$-\frac{12a^5}{125}$$
When the answer should be
$$\frac{4a^5}{125}$$
What have I done wrong?
we have $$(2\frac{a^2}{5})\left(-\frac{2a^3}{25}-\frac{a^3}{25}+\frac{a^3}{5}\right)$$=$$\frac{2a^2}{5}\cdot \frac{2a^3}{25}=$$ $$\frac{4a^5}{125}=0$$ thus $$a=0$$
$$-\frac a5\left(2\times \frac {a^2}5\right)^2=-\frac a5\times \frac {4a^4}{25}=-\frac {4a^5}{125}$$
$$5\left(-\frac a5\right)^3\left(2\times \frac {a^2}5\right)=-\frac {a^3}{25}\left(\frac {2a^2}5\right)=-\frac {2a^5}{125}$$ And $$-a^2\left(-\frac a5\right)\left(2\times \frac {a^2}{5}\right)=\frac {a^3}5\left(2\times\frac {a^2}5\right)=\frac {2a^5}{25}$$
Combining, we get this:$$\color{blue}{-\frac a5\left(\frac {2a^2}5\right)^2}+\color{red}{5\left(-\frac a5\right)^3\left(\frac {2a^2}5\right)}\color{brown}{-a^2\left(-\frac a5\right)\left(\frac {2a^2}5\right)}\\ =\color{blue}{-\frac {4a^5}{125}}\color{red}{-\frac {2a^5}{125}}\color{brown}{+\frac {2a^5}{25}}=-\frac {6a^5}{125}+\frac {2a^5}{25}=\boxed{\frac {4a^5}{125}}$$
For where you went wrong, you put $\frac {-4a^5}{25}$ instead of $-\frac {4a^5}{125}$. Most likely, you forgot to square the denominator of the fraction in $\left(\frac {2a^2}5\right)^2$.
You start with an equation but I suppose you are only concerned about the left hand side of the equation.
$$-\frac{a}{5}\left(2\times\frac{a^2}{5}\right)^2+5\left(-\frac{a}{5}\right)^3\left(2\times\frac{a^2}{5}\right) -a^2\left(-\frac{a}{5}\right)\left(2\times\frac{a^2}{5}\right)\tag{1}$$ I get $$\frac{-4a^5}{25}-\frac{2a^5}{125}+\frac{2a^5}{25}\tag{2}$$ (...) $$-\frac{12a^5}{125}\tag{3}$$
How to find the error?
Plug in some numbers for the variablee $a$ and compare, e.g $a=5$ in $(1)$ will give
$$-\frac{5}{5}\left(2\times\frac{5^2}{5}\right)^2+5\left(-\frac{5}{5}\right)^3\left(2\times\frac{5^2}{5}\right) -5^2\left(-\frac{5}{5}\right)\left(2\times\frac{5^2}{5}\right) \\ =-1(10)^2+5(-1)^3(10)-25(-1)(10)\\ =-100-50+250\\=100$$ but $(2)$ will give $$\frac{-4\cdot 5^5}{25}-\frac{2\cdot 5^5}{125}+\frac{2\cdot 5^5}{25}\\ =-4\cdot 5^3-2\cdot5^2+2\cdot5^3\\ =-500-50+250\\ =300$$ So you introduced an error when you calculated $(2)$ from $(1)$
You can plug in arbitrary numbers for $a$ and use a calculator to verify/disprove your calculation. If a number you plug in verifies your calculation that does not mean that there is no error in the calculation, maybe other values will disprove the calculation. But if a number disproves a calculation you can be sure that there is an error in your calculation and try to find it by using the same method for the substeps of your calculation.
So next you check each term in this step. Ifyou plug in $a=6$ in the first term
$$-\frac{a}{5}\left(2\times\frac{a^2}{5}\right)^2$$
and in the first term of your your result
$$\frac{-4a^5}{25}$$
and use a calculator you will get something like $-248.832$ and $-1244.16$ (You almost always have a calculator. I simply put -6/5*(2*6^2/5)^2 and -4*6^5/25 into Google to get these results).
So there is an error in transforming this term.
You can often use such a technique to find errors in your calculations.
If you have access to a CAS like Mathematica you can use this tool to check your intermediate results.
