https://file.aiquickdraw.com/mathbot/file/17289184796558o72uar4.jpg

Let's work through the multiplication of the polynomials given in the image.

The polynomials are:

\[
P(x) = 3x^2 + 5x - 2
\]
\[
Q(x) = -2x^2 + 4
\]
\[
S(x) = 2x^2
\]
\[
R(x) = 5x - 6
\]

Now, let's perform the requested multiplications.

### 1. \( P(x) \times S(x) \)

\[
P(x) \times S(x) = (3x^2 + 5x - 2) \times 2x^2
\]

We distribute \( 2x^2 \) across each term of \( P(x) \):

\[
= 2x^2(3x^2) + 2x^2(5x) + 2x^2(-2)
\]
\[
= 6x^4 + 10x^3 - 4x^2
\]

Thus, the result is:

\[
P(x) \times S(x) = 6x^4 + 10x^3 - 4x^2
\]

---

### 2. \( Q(x) \times S(x) \)

\[
Q(x) \times S(x) = (-2x^2 + 4) \times 2x^2
\]

We distribute \( 2x^2 \) across each term of \( Q(x) \):

\[
= 2x^2(-2x^2) + 2x^2(4)
\]
\[
= -4x^4 + 8x^2
\]

Thus, the result is:

\[
Q(x) \times S(x) = -4x^4 + 8x^2
\]

---

### 3. \( Q(x) \times R(x) \)

\[
Q(x) \times R(x) = (-2x^2 + 4) \times (5x - 6)
\]

We use the distributive property:

\[
= (-2x^2)(5x) + (-2x^2)(-6) + (4)(5x) + (4)(-6)
\]
\[
= -10x^3 + 12x^2 + 20x - 24
\]

Thus, the result is:

\[
Q(x) \times R(x) = -10x^3 + 12x^2 + 20x - 24
\]

---

### 4. \( P(x) \times Q(x) \)

\[
P(x) \times Q(x) = (3x^2 + 5x - 2) \times (-2x^2 + 4)
\]

We distribute each term of \( P(x) \) over \( Q(x) \):

\[
= (3x^2)(-2x^2) + (3x^2)(4) + (5x)(-2x^2) + (5x)(4) + (-2)(-2x^2) + (-2)(4)
\]
\[
= -6x^4 + 12x^2 - 10x^3 + 20x + 4x^2 - 8
\]

Combining like terms:

\[
= -6x^4 - 10x^3 + 16x^2 + 20x - 8
\]

Thus, the result is:

\[
P(x) \times Q(x) = -6x^4 - 10x^3 + 16x^2 + 20x - 8
\]

---

### 5. \( P(x) \times R(x) \)

\[
P(x) \times R(x) = (3x^2 + 5x - 2) \times (5x - 6)
\]

We distribute each term of \( P(x) \) over \( R(x) \):

\[
= (3x^2)(5x) + (3x^2)(-6) + (5x)(5x) + (5x)(-6) + (-2)(5x) + (-2)(-6)
\]
\[
= 15x^3 - 18x^2 + 25x^2 - 30x - 10x + 12
\]

Combining like terms:

\[
= 15x^3 + 7x^2 - 40x + 12
\]

Thus, the result is:

\[
P(x) \times R(x) = 15x^3 + 7x^2 - 40x + 12
\]

---

Would you like further details or have any questions about these calculations?

Here are five related questions you might find interesting:
1. How do you perform polynomial division for the given polynomials?
2. Can you graph the resulting polynomials from the multiplications?
3. How would the behavior of the functions change if we added higher-degree terms?
4. What are the roots of \(P(x)\), \(Q(x)\), and other polynomials involved?
5. Can you factor the resulting polynomials?

**Tip**: Always simplify polynomials by combining like terms after performing operations like multiplication or addition.

Considera los siguientes polinomios:
P(x) = 3x^2 + 5x - 2
Q(x) = -2x^2 + 4
S(x) = 2x^2
R(x) = 5x - 6

Realiza las siguientes multiplicaciones y expresa el resultado en forma de polinomio ordenado:
1. P(x) * S(x)
2. Q(x) * S(x)
3. Q(x) * R(x)
4. P(x) * Q(x)
5. P(x) * R(x)

Learn how to multiply polynomials step-by-step using the distributive property. This problem involves polynomial expressions P(x) = 3x^2 + 5x - 2, Q(x) = -2x^2 + 4, R(x) = 5x - 6, and S(x) = 2x^2. Follow detailed solutions for each multiplication task, including combining like terms.

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