https://file.aiquickdraw.com/mathbot/file/1729793090171v17dv9k0.jpg

Let's solve the two equations presented in the image step by step.

### Problem 1: Solve the equation
\[
x^3 - 3x^2 - 10x = 0
\]

**Step 1**: Factor the equation.
\[
x(x^2 - 3x - 10) = 0
\]

**Step 2**: Solve \( x = 0 \), which is one solution.

**Step 3**: Solve the quadratic equation \( x^2 - 3x - 10 = 0 \) using the quadratic formula:
\[
x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4(1)(-10)}}{2(1)}
\]
\[
x = \frac{3 \pm \sqrt{9 + 40}}{2} = \frac{3 \pm \sqrt{49}}{2} = \frac{3 \pm 7}{2}
\]

Thus, the two solutions are:
\[
x = \frac{3 + 7}{2} = 5 \quad \text{and} \quad x = \frac{3 - 7}{2} = -2
\]

**Final Solutions** for the first equation:
\[
x = 0, \, x = -2, \, x = 5
\]

---

### Problem 2: Solve the polynomial
\[
4x^3 + 7x^2 = 64x + 112
\]

**Step 1**: Move all terms to one side:
\[
4x^3 + 7x^2 - 64x - 112 = 0
\]

**Step 2**: Factor by grouping. Start by factoring the polynomial:
We try factoring by grouping:
\[
4x^3 - 64x + 7x^2 - 112 = 0
\]

Group terms:
\[
4x(x^2 - 16) + 7(x^2 - 16) = 0
\]
\[
(4x + 7)(x^2 - 16) = 0
\]

**Step 3**: Factor \( x^2 - 16 \) as a difference of squares:
\[
(4x + 7)(x - 4)(x + 4) = 0
\]

**Step 4**: Solve for \( x \):
\[
x = -\frac{7}{4}, \quad x = 4, \quad x = -4
\]

**Final Solutions** for the second equation:
\[
x = -\frac{7}{4}, \, x = -4, \, x = 4
\]

---

Would you like further details on any of the steps? 

Here are five related questions to practice:

1. How do you factor a cubic equation by grouping?
2. What is the quadratic formula, and how is it applied?
3. How can you check if your solutions are correct by plugging them back into the original equation?
4. Can every cubic equation be factored by grouping?
5. How do you identify when to use the difference of squares in factoring?

**Tip**: When solving polynomials, always check if there's a common factor or if grouping can simplify the problem!

Solve the cubic equation: x^3 - 3x^2 - 10x = 0 and Solve the polynomial equation: 4x^3 + 7x^2 = 64x + 112.

Step-by-step solution to solving the cubic equation x^3 - 3x^2 - 10x = 0 and the polynomial equation 4x^3 + 7x^2 = 64x + 112. Learn key algebraic techniques including factoring by grouping, the quadratic formula, and the difference of squares. Suitable for students in Grades 9-12.

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