https://r2.erweima.ai/mathbot/image/1725643544205tmvm1yrv.jpg

The image has been loaded successfully. I'll now analyze the mathematical statements to determine whether each is true or false.

Let's go through each part:

**a.** \((\sqrt{x} - 3)^2 = x - 9\)

Expand the left-hand side:
\[
(\sqrt{x} - 3)^2 = (\sqrt{x})^2 - 2(\sqrt{x})(3) + 9 = x - 6\sqrt{x} + 9
\]
Clearly, \(x - 6\sqrt{x} + 9 \neq x - 9\), so **a** is false.

**b.** \((\sqrt{x} + 3)(\sqrt{x} - 3) = x - 9\)

This is a difference of squares:
\[
(\sqrt{x} + 3)(\sqrt{x} - 3) = (\sqrt{x})^2 - 3^2 = x - 9
\]
Thus, **b** is true.

**c.** \(2(2x - 1)(x + 1) = (2x - 2)(2x + 2)\)

First, expand both sides.

Left-hand side:
\[
2(2x - 1)(x + 1) = 2 \times (2x^2 + 2x - x - 1) = 4x^2 + 2x - 2
\]

Right-hand side:
\[
(2x - 2)(2x + 2) = 4x^2 + 4x - 4x - 4 = 4x^2 - 4
\]
Clearly, \(4x^2 + 2x - 2 \neq 4x^2 - 4\), so **c** is false.

**d.** \((x^2 - 3)^2 = x^2 - 6\sqrt{x} + 9\)

Expand the left-hand side:
\[
(x^2 - 3)^2 = x^4 - 6x^2 + 9
\]
Clearly, \(x^4 - 6x^2 + 9 \neq x^2 - 6\sqrt{x} + 9\), so **d** is false.

**e.** The expression \(p(x) = 4x^3 - x^2 + 4\) is a polynomial.

Yes, this is a cubic polynomial, so **e** is true.

**f.** The degree of \(x^2 - x^3 + 1)(x - 1) = 4\)

The degree of \((x^2 - x^3 + 1)\) is 3, and the degree of \((x - 1)\) is 1. Therefore, the degree of the product is \(3 + 1 = 4\), so **f** is true.

**g.** \(\sqrt{x} + \sqrt{x} = 2x\)

This is not true, since \(\sqrt{x} + \sqrt{x} = 2\sqrt{x}\), not \(2x\), so **g** is false.

**h.** The expression \((m + n)^3 - (m + n)(m - n)\) is equal to \(2mn\).

Let's expand both sides.

Left-hand side:
\[
(m + n)^3 - (m + n)(m - n) = m^3 + 3m^2n + 3mn^2 + n^3 - (m^2 - n^2) = m^3 + 3m^2n + 3mn^2 + n^3 - m^2 + n^2
\]
This does not simplify to \(2mn\), so **h** is false.

### Summary:
a. F  
b. V  
c. F  
d. F  
e. V  
f. V  
g. F  
h. F

Analizar la verdad o falsedad de cada uno de los anunciados. Justificar.

Analyze and determine the truth or falsehood of various mathematical statements involving algebra, polynomial expressions, square roots, and degree of polynomials. Detailed explanations for each statement are provided.

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