What the answer of an oil painting is 10 in . longer than its wide and is bordered on all sides by a 3 in.. wide frame . its area of the frame alone is 420 in^2 ,what is the dimensions of the painting]

You know that one side is 10 inches longer than the other, but don't know any other information regarding length so you can represent the side lengths as (x) and (x + 10) 

You also know that there is 3 addition inches on each side from the frame so for the entire area you can represent the sides as (x + 3) and (x + 13)

The area of the painting alone is the entire area minus the area of the frame, which is given 

If the entire area is given by (x + 3) * (x + 13)
then you can use subtract the frame area from this as (x + 3) * (x + 13) - 420

The best way to begin solving this is to form an equation:
(x + 3) * (x + 13) - 420 = 0

and then rearrange it to (x + 3) * (x +13) = 420 
(I did this by adding 420 to both sides of the equation) 

Now, expand the left side of the equation with the distributive property, which gives you x² + 13x + 3x + 39 

Simplified to x² + 16x + 39 = 420 

Now, you can move the 420 back to the other side (by subtracting the 420 from each side) to get the quadratic function:

x² + 16x - 381

Back to the painting, you can now find the roots (x intercepts) of the quadratic function in order to find the value of x in the painting length. 

Hope this helps :)