\begin{tabular}{l} 4. Simplify: \ a ) $$ y^{2}-(-y)^{2} $$ \ b) $$ 2 x(3 x)-3 x(-2 x) $$ \ c) $$ -a(a+1)-2(3-a)+a(a $$ \ d) $$ -4 m\left(2 n^{2} ight) \div(-2 n)^{2}+3 m $$ \ e ) $$ 3 b(2 c-3 d)-6 c(b+d)+ $$ \ f) $$ 3 x y(2-x)-2 y(3 x+1)-y $$ \ \hline\end{tabular}

Question

\begin{tabular}{l} 4. Simplify: \ a ) $$ y^{2}-(-y)^{2} $$ \ b) $$ 2 x(3 x)-3 x(-2 x) $$ \ c) $$ -a(a+1)-2(3-a)+a(a $$ \ d) $$ -4 m\left(2 n^{2}ight) \div(-2 n)^{2}+3 m $$ \ e ) $$ 3 b(2 c-3 d)-6 c(b+d)+ $$ \ f) $$ 3 x y(2-x)-2 y(3 x+1)-y $$ \ \hline\end{tabular}

UpStudy AI · Accepted Answer

Simplify the expression by following steps:
- step0: Solution:
  $$3xy\left(2-x\right)-2y\left(3x+1\right)-y$$
- step1: Expand the expression:
  $$6xy-3x^{2}y-2y\left(3x+1\right)-y$$
- step2: Expand the expression:
  $$6xy-3x^{2}y-6yx-2y-y$$
- step3: Subtract the terms:
  $$0-3x^{2}y-2y-y$$
- step4: Remove 0:
  $$-3x^{2}y-2y-y$$
- step5: Subtract the terms:
  $$-3x^{2}y-3y$$
Calculate or simplify the expression $$ y^{2}-(-y)^{2} $$.
Simplify the expression by following steps:
- step0: Solution:
  $$y^{2}-\left(-y\right)^{2}$$
- step1: Rewrite the expression:
  $$y^{2}-y^{2}$$
- step2: Subtract the terms:
  $$0$$
Calculate or simplify the expression $$ 2 x(3 x)-3 x(-2 x) $$.
Simplify the expression by following steps:
- step0: Solution:
  $$2x\times 3x-3x\left(-2x\right)$$
- step1: Multiply:
  $$6x^{2}-3x\left(-2x\right)$$
- step2: Multiply:
  $$6x^{2}+6x^{2}$$
- step3: Collect like terms:
  $$\left(6+6\right)x^{2}$$
- step4: Add the numbers:
  $$12x^{2}$$
Let's simplify each expression step by step:

### a) $$ y^{2}-(-y)^{2} $$

1. Calculate $$ (-y)^{2} $$:
   $$
   (-y)^{2} = y^{2}
   $$
2. Substitute back into the expression:
   $$
   y^{2} - y^{2} = 0
   $$

**Result:** $$ 0 $$

---

### b) $$ 2 x(3 x)-3 x(-2 x) $$

1. Calculate $$ 2 x(3 x) $$:
   $$
   2 x(3 x) = 6 x^{2}
   $$
2. Calculate $$ -3 x(-2 x) $$:
   $$
   -3 x(-2 x) = 6 x^{2}
   $$
3. Substitute back into the expression:
   $$
   6 x^{2} + 6 x^{2} = 12 x^{2}
   $$

**Result:** $$ 12 x^{2} $$

---

### c) $$ -a(a+1)-2(3-a)+a(a) $$

1. Expand $$ -a(a+1) $$:
   $$
   -a^{2} - a
   $$
2. Expand $$ -2(3-a) $$:
   $$
   -6 + 2a
   $$
3. Expand $$ a(a) $$:
   $$
   a^{2}
   $$
4. Combine all parts:
   $$
   -a^{2} - a - 6 + 2a + a^{2} = a - 6
   $$

**Result:** $$ a - 6 $$

---

### d) $$ -4 m(2 n^{2}) \div (-2 n)^{2}+3 m $$

1. Calculate $$ (-2 n)^{2} $$:
   $$
   (-2 n)^{2} = 4 n^{2}
   $$
2. Calculate $$ -4 m(2 n^{2}) \div 4 n^{2} $$:
   $$
   -4 m(2 n^{2}) \div 4 n^{2} = -2m
   $$
3. Combine with $$ 3 m $$:
   $$
   -2m + 3m = m
   $$

**Result:** $$ m $$

---

### e) $$ 3 b(2 c-3 d)-6 c(b+d) $$

1. Expand $$ 3 b(2 c-3 d) $$:
   $$
   6 bc - 9 bd
   $$
2. Expand $$ -6 c(b+d) $$:
   $$
   -6 bc - 6 cd
   $$
3. Combine all parts:
   $$
   6 bc - 9 bd - 6 bc - 6 cd = -9 bd - 6 cd
   $$

**Result:** $$ -9 bd - 6 cd $$

---

### f) $$ 3 x y(2-x)-2 y(3 x+1)-y $$

1. Expand $$ 3 x y(2-x) $$:
   $$
   6xy - 3x^{2}y
   $$
2. Expand $$ -2 y(3 x+1) $$:
   $$
   -6xy - 2y
   $$
3. Combine all parts:
   $$
   6xy - 3x^{2}y - 6xy - 2y - y = -3x^{2}y - 3y
   $$

**Result:** $$ -3x^{2}y - 3y $$

---

### Summary of Results:
- a) $$ 0 $$
- b) $$ 12 x^{2} $$
- c) $$ a - 6 $$
- d) $$ m $$
- e) $$ -9 bd - 6 cd $$
- f) $$ -3x^{2}y - 3y $$
a) $$ 0 $$ b) $$ 12x^{2} $$ c) $$ a - 6 $$ d) $$ m $$ e) $$ -9bd - 6cd $$ f) $$ -3x^{2}y - 3y $$

\begin{tabular}{l} 4. Simplify: \\ a ) \( y^{2}-(-y)^{2} \) \\ b) \( 2 x(3 x)-3 x(-2 x) \) \\ c) \( -a(a+1)-2(3-a)+a(a \) \\ d) \( -4 m\left(2 n^{2}\right) \div(-2 n)^{2}+3 m \) \\ e ) \( 3 b(2 c-3 d)-6 c(b+d)+ \) \\ f) \( 3 x y(2-x)-2 y(3 x+1)-y \) \\ \hline\end{tabular}

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