Recognize and Use the Appropriate Method to Factor a Polynomial Completely.

$10m^4-6250$.

We have $10m^4-6250$.

We will first take out the common factors from the equation.

The we will get to see a pattern of difference of two perfect squares, i.e., ${\left(a\right)}^2-{\left(b\right)}^2=(a+b)(a-b)$.

By applying this, we will get the factors of the equation.

We have $10m^4-6250$.

Taking $10$common gives us,

$10(m^4-625)$.

We know that${\left(m^2\right)}^2=m^4$ and
${\left(25\right)}^2=625$.

We have $10(m^4-625)$.

We know that they are perfect squares. So applying the property of difference of two perfect squares, we get

$=10(m^2+25)(m^2-25)$

We can further apply the property in $(m^2-25)$as${\left(m\right)}^2=m^2$ and
${\left(5\right)}^2=25$.

So by applying he property we get,

$=10(m^2+25)(m+5)(m-5)$

This cannot be further factored so they are the factors of the expression.

We will check the solution by simply multiplying the factors. If we get the given equation as the product, our answer is right.

So,

$$\begin{gathered} =10(m^2+25)(m+5)(m-5) \\ =\left[10m^2+250\right]\left[m^2-25\right] \\ =10m^4-250m^2+250m^2-6250 \\ =10m^4-6250 \end{gathered}$$

Thus our calculations are right.