Not really up to the factoring problems, I'm tired and haven't had factoring math in quite some time so hopefully someone else helps you with that, I can help with the first one though.
Basically you come up with equations to represent the length and width in terms of each other.
If your length is L and your Width is W, the length can be represented as W + 15 because it is 15 yards longer than the width right? The second equation is just the equation for area which means L x W = 450 square yards. If you have two equations and two variables you can use several different methods to identify those variables so choose your favorite, I will be using substitution because it's easy.
We need to literally find one variable in terms of another, luckily we already have. L = W + 15. If we then substitute all L's in the second equation for W's instead we will have an equation with only one variable which we can solve for and then replace in the first equation to solve for L.
Second equation with L's substituted with W + 15 instead.
W(W+15) = 450
W^2 +15W = 450
W^2 +15W - 450 = 0
Now we have a nice little quadratic equation that we can solve for W, lets do factoring.
Factoring is taking a large polynomial and finding smaller binomials that multiply to get the original large polynomial. When dealing with quadratics theres a few tricks to getting these values without making things too complicated. When multiplying two binomials like (x + a)(x + b) you always end up with x^2 + ax + bx + ab with the x terms combining because a and b are just numbers.
This means that if you can find two numbers that add to get the middle term and multiply to get the 3rd term you just found your binomials easy as that.
In our case we need two numbers that add to get 15 and multiply to get -450. Thinking about if for a second we can try 45 x -10 but those don't add to get 15
We can try 30 x -15 which is correct. 30 x -15 = -450 30 - 15 = 15
Now we write the binomials so instead of the original W^2 + 15W - 450 we have
(W+30)(W-15) = 0
Now we can find the "roots" which just means what value of W is needed to make this equation true? If either of the binomials equals 0 the whole thing will equal 0 so our roots are -30 and 15. Since length cannot be negative, our W must equal 15 yards.
Now for a much simpler calculation, our original first equation L = W + 15 which means L = 15 + 15 which means our length is 30.
Width = 15 yards Length = 30 yards.
Hope that helped
I am receiving my 4-year degree in Chemistry with a minor in Mathematics in May of next year.