Let's solve the polynomial $-25x + 50 + x^3 - 2x^2 = 0$ and express it in factored form as requested in the image.

Step 1: Rewrite in Standard Form

The given polynomial is already close to standard form but we can rearrange it as: $x^3 - 2x^2 - 25x + 50 = 0$ This is the standard form of the cubic equation.

Step 2: Use Synthetic Division to Find Roots

To use synthetic division, we first try to find possible rational roots using the Rational Root Theorem. The possible rational roots are the factors of the constant term (50) divided by the factors of the leading coefficient (1). So, the possible roots are: $\pm 1, \pm 2, \pm 5, \pm 10, \pm 25, \pm 50$

Let's try some values by substituting them into the polynomial:

• For $x = 2$: $(2)^3 - 2(2)^2 - 25(2) + 50 = 8 - 8 - 50 + 50 = 0$ So, $x = 2$ is a root.

Step 3: Perform Synthetic Division

We divide $x^3 - 2x^2 - 25x + 50$ by $x - 2$.

The coefficients of the polynomial are: $1, -2, -25, 50$.

Performing synthetic division: