https://file.aiquickdraw.com/mathbot/file/17274114226902zxqfemt.jpg

Let's go through the problems in the image and solve them one by one.

### Problem g:
\[ 3m^2 - [(2m^2 - m + 1)(m^2 + m) - \{2m(m^2 + 1)\}] \]

Step 1: Expand both terms inside the brackets:
- For \((2m^2 - m + 1)(m^2 + m)\):
  \[
  = 2m^2(m^2 + m) - m(m^2 + m) + 1(m^2 + m)
  = 2m^4 + 2m^3 - m^3 - m^2 + m^2 + m
  = 2m^4 + m^3 + m
  \]

- For \(2m(m^2 + 1)\):
  \[
  = 2m(m^2 + 1) = 2m^3 + 2m
  \]

Step 2: Subtract the two results:
\[
(2m^4 + m^3 + m) - (2m^3 + 2m) = 2m^4 - m^3 - m
\]

Step 3: Substitute back into the expression:
\[
3m^2 - (2m^4 - m^3 - m)
= 3m^2 - 2m^4 + m^3 + m
\]

So the simplified expression is:
\[
\boxed{-2m^4 + m^3 + 3m^2 + m}
\]

---

### Problem h:
\[ 6k^2 - [(2k^2 - k)(k^2 + k) + \{(k^2 + k)(k + 3)\}] \]

Step 1: Expand the terms inside the brackets:
- For \((2k^2 - k)(k^2 + k)\):
  \[
  = 2k^2(k^2 + k) - k(k^2 + k)
  = 2k^4 + 2k^3 - k^3 - k^2
  = 2k^4 + k^3 - k^2
  \]

- For \((k^2 + k)(k + 3)\):
  \[
  = k^2(k + 3) + k(k + 3)
  = k^3 + 3k^2 + k^2 + 3k
  = k^3 + 4k^2 + 3k
  \]

Step 2: Add the two results:
\[
(2k^4 + k^3 - k^2) + (k^3 + 4k^2 + 3k) = 2k^4 + 2k^3 + 3k^2 + 3k
\]

Step 3: Substitute back into the expression:
\[
6k^2 - (2k^4 + 2k^3 + 3k^2 + 3k)
= 6k^2 - 2k^4 - 2k^3 - 3k^2 - 3k
= -2k^4 - 2k^3 + 3k^2 - 3k
\]

So the simplified expression is:
\[
\boxed{-2k^4 - 2k^3 + 3k^2 - 3k}
\]

---

Would you like me to continue with the other problems (i, j, k, l), or do you have any questions so far? Feel free to ask for clarification!

Here are 5 related questions to practice:
1. How would you simplify an expression with nested parentheses and variables like \( a(a^2 + b) - b(b^2 + a) \)?
2. How does distributing work when multiplying two binomials, such as \( (x + 2)(x - 3) \)?
3. Can you solve \( 5m^3 - (3m^2 - m) \) by combining like terms?
4. What strategies help in recognizing and simplifying higher-degree polynomials?
5. How can factoring simplify polynomial subtraction or addition?

### Tip:
When solving expressions with nested brackets or parentheses, always start simplifying the innermost parts first, then work outward to avoid mistakes.

Solve the following algebraic expressions: g) 3m^2 - [(2m^2 - m + 1)(m^2 + m) - {2m(m^2 + 1)}], h) 6k^2 - [(2k^2 - k)(k^2 + k) + {(k^2 + k)(k + 3)}], i) 8u^2 - [(2u^2 - u)(u^2 + u + 3) - 6u^2(u^3 + 1)], j) 21v^2 - [(2v^2(v^2 + v) - {4v^2(v + 1)}], k) (64x - (12x - 5x)) ÷ 3, l) (12x^2 + 5x^2) ÷ 4x.

This page provides solutions to algebraic expression problems, focusing on polynomial expansion and simplification techniques. Learn how to apply the distributive property and simplify complex algebraic terms. Suitable for students in Grades 9-11.

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