Factor: f(x)=2x^4-5x^3-30x^2+80x-32 a) f(x)=(2x-1)(x-2)(x+4)(x-4) b) f(x)=(2x-1)(x-2)(x-4)^2 c) f(x)=(2x+1)(x-2)(x-4)^2 d) f(x)=(2x+1)(x+2)(x-4)^2

## Step 1: Rational Root Theorem### The Rational Root Theorem helps identify potential rational roots of a polynomial. In this case, potential rational roots are of the form , where is a factor of the constant term (-32) and is a factor of the leading coefficient (2). Possible rational roots are .## Step 2: Testing Potential Roots### Substitute the potential roots into the polynomial . We find that . Therefore, is a root, and is a factor.## Step 3: Polynomial Division### Divide by using synthetic division or polynomial long division. This yields .## Step 4: Factoring the Cubic### Now we factor the cubic . Testing potential roots again, we find that is a root since . Thus, is a factor.## Step 5: Further Polynomial Division### Dividing by gives .## Step 6: Difference of Squares### is a difference of squares, factoring to .## Step 7: Final Factorization### Combining all factors, we have .