Factor each expression completely. Can you please show me the steps on how you reached your answer so I know how to these types of problems thank you.

3x^4+13x^3y-10x^2y^2

First look for common factors

3x^4+13x^3y-10x^2y^2
= x^2(3x^2 + 13xy - 10y^2)
= x^2(3x -2y)(x + 5y)

(you should have your own method of factoring the above quadratic)

To factor the expression 3x^4 + 13x^3y - 10x^2y^2 completely, we will look for common factors and use factoring techniques.

Step 1: Identify common factors
First, we look for common factors among the terms in the expression. In this case, the terms share a common factor of x^2:

3x^4 = x^2(3x^2)
13x^3y = x^2(13xy)
-10x^2y^2 = x^2(-10y^2)

Step 2: Factor out the common factor
Next, we factor out the common factor, which is x^2 in this case:

3x^4 + 13x^3y - 10x^2y^2 = x^2(3x^2 + 13xy - 10y^2)

Now, we have factored out the common factor x^2 from each term.

Step 3: Factor the remaining quadratic expression
The expression within the parentheses, 3x^2 + 13xy - 10y^2, is a quadratic expression. We need to factor it further.

To factor this quadratic expression, we look for two numbers that multiply to give the product of the coefficient of the x^2 term (3) and the constant term (-10), which is -30, and that add up to the coefficient of the linear term (13).

The numbers that satisfy these conditions are 15 and -2, because:
15 * (-2) = -30 (product)
15 + (-2) = 13 (sum)

Therefore, we can rewrite the quadratic expression as follows:

3x^2 + 13xy - 10y^2 = (3x^2 + 15xy) + (-2xy - 10y^2)

Step 4: Group terms and factor by grouping
Now, we group the terms and factor by grouping:

= 3x^2( x + 5y) - 2y(x + 5y)

Step 5: Factor out the common binomial factor
Finally, we factor out the common binomial factor, (x + 5y):

= (x + 5y)(3x^2 - 2y)

So, the completely factored form of the expression 3x^4 + 13x^3y - 10x^2y^2 is (x + 5y)(3x^2 - 2y).