1. Problem statement: Factorize the given expressions: - $C = x^2 - 9$ - $D = (4 + x)(2x - 1) + 2(4 + x)$ - $E = 49x^2 + 14x + 1 + 3(7x + 1)$ - $F = (\sqrt{2}x - 1)(x - 3) + \left(x - \frac{1}{\sqrt{2}}\right)(x - 1)$ --- 2. Factorize $C = x^2 - 9$: This is a difference of squares: $$x^2 - 9 = (x)^2 - (3)^2 = (x - 3)(x + 3)$$ --- 3. Factorize $D = (4 + x)(2x - 1) + 2(4 + x)$: Factor $(4 + x)$ from both terms: $$D = (4 + x)(2x - 1) + 2(4 + x) = (4 + x)((2x - 1) + 2)$$ Simplify inside the parenthesis: $$(2x - 1) + 2 = 2x + 1$$ So: $$D = (4 + x)(2x + 1)$$ --- 4. Factorize $E = 49x^2 + 14x + 1 + 3(7x + 1)$: First, expand the last term: $$3(7x + 1) = 21x + 3$$ Add this to the rest: $$E = 49x^2 + 14x + 1 + 21x + 3 = 49x^2 + (14x + 21x) + (1 + 3) = 49x^2 + 35x + 4$$ Now, factor $E = 49x^2 + 35x + 4$. Look for two numbers whose product is $49 \times 4 = 196$ and sum is $35$. These numbers are $28$ and $7$ because $28 \times 7 = 196$ and $28 + 7 = 35$. Rewrite: $$49x^2 + 28x + 7x + 4$$ Group: $$(49x^2 + 28x) + (7x + 4)$$ Factor each group: $$7x(7x + 4) + 1(7x + 4)$$ Factor out common term $(7x + 4)$: $$(7x + 4)(7x + 1)$$ So, $$E = (7x + 4)(7x + 1)$$ --- 5. Factorize $F = (\sqrt{2}x - 1)(x - 3) + \left(x - \frac{1}{\sqrt{2}}\right)(x - 1)$: First, expand each product: $$ (\sqrt{2}x - 1)(x - 3) = \sqrt{2}x^2 - 3\sqrt{2}x - x + 3 $$ $$ \left(x - \frac{1}{\sqrt{2}}\right)(x - 1) = x^2 - x - \frac{x}{\sqrt{2}} + \frac{1}{\sqrt{2}} $$ Now sum these: $$ F = (\sqrt{2}x^2 - 3\sqrt{2}x - x + 3) + (x^2 - x - \frac{x}{\sqrt{2}} + \frac{1}{\sqrt{2}}) $$ Group like terms: Quadratic terms: $$ \sqrt{2}x^2 + x^2 = (\sqrt{2} + 1)x^2 $$ Linear terms: $$ -3\sqrt{2}x - x - x - \frac{x}{\sqrt{2}} = -3\sqrt{2}x - 2x - \frac{x}{\sqrt{2}} $$ Constants: $$ 3 + \frac{1}{\sqrt{2}} $$ Rewrite linear terms with common factor x: $$ x \left(-3\sqrt{2} - 2 - \frac{1}{\sqrt{2}}\right) $$ So, $$ F = (\sqrt{2} + 1)x^2 + x \left(-3\sqrt{2} - 2 - \frac{1}{\sqrt{2}}\right) + 3 + \frac{1}{\sqrt{2}} $$ This expression does not simplify nicely by common factoring. Hence, $F$ is simplified as the above expression. --- Final factorized forms: - $C = (x - 3)(x + 3)$ - $D = (4 + x)(2x + 1)$ - $E = (7x + 4)(7x + 1)$ - $F = (\sqrt{2} + 1)x^2 + x \left(-3\sqrt{2} - 2 - \frac{1}{\sqrt{2}}\right) + 3 + \frac{1}{\sqrt{2}}$