1. **Factorize each expression completely:** (1) Factorize $x^2 - 1$: - This is a difference of squares: $a^2 - b^2 = (a-b)(a+b)$. - Here, $a = x$, $b = 1$. - So, $x^2 - 1 = (x-1)(x+1)$. (2) Factorize $16x^2 - 9$: - This is also a difference of squares: $16x^2 = (4x)^2$, $9 = 3^2$. - So, $16x^2 - 9 = (4x - 3)(4x + 3)$. (3) Factorize $x^3 + 1$: - This is a sum of cubes: $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$. - Here, $a = x$, $b = 1$. - So, $x^3 + 1 = (x + 1)(x^2 - x + 1)$. (4) Factorize $27 - a^3 b^3$: - Rewrite as $27 - (ab)^3$. - This is a difference of cubes: $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$. - Here, $a = 3$, $b = ab$. - So, $27 - a^3 b^3 = (3 - ab)(9 + 3ab + a^2 b^2)$. (5) Factorize $4x^4 + 4000x$: - Factor out the greatest common factor (GCF): $4x$. - $4x^4 + 4000x = 4x(x^3 + 1000)$. - Recognize $x^3 + 1000$ as sum of cubes: $x^3 + 10^3$. - So, $x^3 + 1000 = (x + 10)(x^2 - 10x + 100)$. - Final factorization: $4x(x + 10)(x^2 - 10x + 100)$. 2. **Given:** $4x^2 - 9y^2 = 115$ and $2x - 3y = 5$. Find the value of $2x + 3y$. - Note that $4x^2 - 9y^2 = (2x)^2 - (3y)^2$ is a difference of squares. - So, $4x^2 - 9y^2 = (2x - 3y)(2x + 3y)$. - Substitute $2x - 3y = 5$: $$115 = 5 imes (2x + 3y)$$ - Divide both sides by 5: $$2x + 3y = \frac{115}{5} = 23$$ 3. **Solve the equation in $\mathbb{R}$:** $$x^2 + 6x = -9$$ - Move all terms to one side: $$x^2 + 6x + 9 = 0$$ - Recognize this as a perfect square trinomial: $$x^2 + 6x + 9 = (x + 3)^2$$ - So, $(x + 3)^2 = 0$ implies $x + 3 = 0$. - Therefore, $x = -3$. **Final answers:** (1) $(x-1)(x+1)$ (2) $(4x-3)(4x+3)$ (3) $(x+1)(x^2 - x + 1)$ (4) $(3 - ab)(9 + 3ab + a^2 b^2)$ (5) $4x(x + 10)(x^2 - 10x + 100)$ Value of $2x + 3y$ is $23$. Solution set of $x^2 + 6x = -9$ is $\{ -3 \}$.