Escribe [tex]$V$[/tex] si el enunciado es verdadero o [tex]$F$[/tex] si es falso.

a. Al resolver la multiplicación de [tex]$\left(7x^2-12\right)\left(x^3+2x+5\right)$[/tex] se obtiene como resultado [tex]$7x^5 + 2x^3 + 35x^2 - 24x - 60$[/tex].

b. El producto de los polinomios [tex]$(x+4)\left(x^2+x+3\right)$[/tex] es [tex]$4x^3 - 5x^2 - 17x + 12$[/tex].

c. Al resolver la multiplicación de [tex]$\left(2x^2+8\right)\left(7x^2+9x-5\right)$[/tex] se obtiene como resultado [tex]$14x^4 - 18x^3 - 46x^2 - 72x + 40$[/tex].

d. El producto de los polinomios [tex]$\left(x^2+3\right)\left(x^4-2x^2+9\right)$[/tex] es [tex]$x^6 + x^4 + 3x^2 + 27$[/tex].

e. [tex]$(4x-6)(8x-2) = 32x^2 + 12$[/tex].

Sure! Let's go through each part of the problem and determine whether each expression results in the given polynomial when the expressions are multiplied.

### h. Expression: [tex]$(4x - 6)(8x - 2)$[/tex]

To verify:
- Multiply the terms: [tex]$(4x)(8x) + (4x)(-2) + (-6)(8x) + (-6)(-2)$[/tex]
- Simplify: [tex]$32x^2 - 8x - 48x + 12$[/tex]
- Combine like terms: [tex]$32x^2 - 56x + 12$[/tex]

Check if [tex]$32x^2 + 12$[/tex] matches: No, it doesn't. So, it's F (False).

### a. Expression: [tex]$(7x^2 - 12)(x^3 + 2x + 5)$[/tex]

To verify:
- Multiply each term of the first polynomial by each term of the second polynomial.
- This results in:
[tex]\[
7x^2 \cdot x^3 + 7x^2 \cdot 2x + 7x^2 \cdot 5 + (-12) \cdot x^3 + (-12) \cdot 2x + (-12) \cdot 5
\][/tex]
- Simplify and combine like terms:
[tex]\[
7x^5 + 14x^3 + 35x^2 - 12x^3 - 24x - 60
\][/tex]
- Combine like terms:
[tex]\[
7x^5 + 2x^3 + 35x^2 - 24x - 60
\][/tex]

Check if it matches with the given polynomial: Yes, it matches. So, it's V (True).

### b. Expression: [tex]$(x + 4)(x^2 + x + 3)$[/tex]

To verify:
- Multiply the terms:
[tex]\[
(x)(x^2) + (x)(x) + (x)(3) + (4)(x^2) + (4)(x) + (4)(3)
\][/tex]
- Simplify:
[tex]\[
x^3 + x^2 + 3x + 4x^2 + 4x + 12
\][/tex]
- Combine like terms:
[tex]\[
x^3 + 5x^2 + 7x + 12
\][/tex]

Check if [tex]$4x^3 - 5x^2 - 17x + 12$[/tex] matches: No, it doesn't. So, it's F (False).

### c. Expression: [tex]$(2x^2 + 8)(7x^2 + 9x - 5)$[/tex]

To verify:
- Multiply each term:
[tex]\[
(2x^2)(7x^2) + (2x^2)(9x) + (2x^2)(-5) + (8)(7x^2) + (8)(9x) + (8)(-5)
\][/tex]
- Simplify:
[tex]\[
14x^4 + 18x^3 - 10x^2 + 56x^2 + 72x - 40
\][/tex]
- Combine like terms:
[tex]\[
14x^4 + 18x^3 + 46x^2 + 72x - 40
\][/tex]

Check if [tex]$14x^4 - 18x^3 - 46x^2 - 72x + 40$[/tex] matches: No, it doesn't. So, it's F (False).

### d. Expression: [tex]$(x^2 + 3)(x^4 - 2x^2 + 9)$[/tex]

To verify:
- Multiply each term:
[tex]\[
x^2 \cdot x^4 + x^2 \cdot -2x^2 + x^2 \cdot 9 + 3 \cdot x^4 + 3 \cdot -2x^2 + 3 \cdot 9
\][/tex]
- Simplify:
[tex]\[
x^6 - 2x^4 + 9x^2 + 3x^4 - 6x^2 + 27
\][/tex]
- Combine like terms:
[tex]\[
x^6 + x^4 + 3x^2 + 27
\][/tex]

Check if it matches the given polynomial: Yes, it matches. So, it's V (True).

The complete results for the statements are: F (False), V (True), F (False), F (False), V (True).