https://file.aiquickdraw.com/mathbot/file/1725654240846kep3jy2f.jpg

I see you've uploaded an image of some math problems. Let me transcribe the relevant mathematical content from the image for analysis.

Here are the questions as seen in the image:

1. **Question 51 (1 point)**:
   \[
   ab^2 + abc^2 = 2abc^2
   \]
   - The possible answer options appear to be missing or unclear in the image.

2. **Question 55 (1 point)**:
   \[
   10m^2 - 10m = 20m^3
   \]
   - Again, the possible answers are unclear.

3. **Question 50 (1 point)**:
   \[
   2x^5 + 3x^4y = y(2x^4 + 3x^3y)
   \]

### Solutions:

1. **For Question 51**:
   Simplifying the equation \( ab^2 + abc^2 = 2abc^2 \):
   \[
   ab^2 + abc^2 = ab(b + c^2)
   \]
   This expression doesn't simplify directly to the given right-hand side \( 2abc^2 \), so this equation might not be true under normal circumstances unless specific values for variables are provided.

2. **For Question 55**:
   The equation is \( 10m^2 - 10m = 20m^3 \). Rearranging:
   \[
   10m^2 - 10m - 20m^3 = 0
   \]
   Factor out the common term:
   \[
   10m(m - 1 - 2m^2) = 0
   \]
   The solutions are either \( m = 0 \) or solving \( m - 1 - 2m^2 = 0 \), which is a quadratic equation.

3. **For Question 50**:
   The equation \( 2x^5 + 3x^4y = y(2x^4 + 3x^3y) \):
   Expanding the right-hand side:
   \[
   y(2x^4 + 3x^3y) = 2x^4y + 3x^3y^2
   \]
   The two sides are not equal unless \( y = x \), which would result in identical terms on both sides.

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Would you like more detailed steps for any of these solutions, or do you have other questions? Here are some related practice questions:

1. Solve for \( m \) in \( 5m^3 - 10m^2 + m = 0 \).
2. Simplify \( x^2 + xy^2 - x^2y \).
3. Factor \( ab^2 + ac^2 - ab \).
4. Solve \( 2x^2 + 3x = 4x^3 \).
5. Solve for \( x \) in \( 2x^3 + 3x^2 = x(2x^2 + 3x) \).

**Tip**: When simplifying equations, always look for common factors that can be factored out before applying more complex operations.

Solve the following equations: (1) ab^2 + abc^2 = 2abc^2, (2) 10m^2 - 10m = 20m^3, (3) 2x^5 + 3x^4y = y(2x^4 + 3x^3y)

This problem involves simplifying and solving polynomial equations. It covers algebraic concepts such as factoring common terms and solving quadratic equations using the zero product property. Ideal for students in Grades 9-12.

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