QUESTION 2 Factorize the following completely: \( 2.1 \quad x^{4}-49 \) \( 2.2 \quad 20-5 x^{2} \) \( 2.3 \quad x^{2}-8 x+15 \) \( 24-x^{3}-8^{3} \) \( 25 a^{2}\left(x^{2}+5 x-6\right) d t b^{2}\left(6-5 x-x^{2}\right) \)

Factor the expression by following steps: - step0: Factor: \(x^{2}-8x+15\) - step1: Rewrite the expression: \(x^{2}+\left(-3-5\right)x+15\) - step2: Calculate: \(x^{2}-3x-5x+15\) - step3: Rewrite the expression: \(x\times x-x\times 3-5x+5\times 3\) - step4: Factor the expression: \(x\left(x-3\right)-5\left(x-3\right)\) - step5: Factor the expression: \(\left(x-5\right)\left(x-3\right)\) Factor the expression \( 20-5 x^{2} \). Factor the expression by following steps: - step0: Factor: \(20-5x^{2}\) - step1: Factor the expression: \(5\left(4-x^{2}\right)\) - step2: Factor the expression: \(5\left(2-x\right)\left(2+x\right)\) Factor the expression \( x^{4}-49 \). Factor the expression by following steps: - step0: Factor: \(x^{4}-49\) - step1: Rewrite the expression: \(\left(x^{2}\right)^{2}-7^{2}\) - step2: Factor the expression: \(\left(x^{2}-7\right)\left(x^{2}+7\right)\) Factor the expression \( 25 a^{2}\left(x^{2}+5 x-6\right) d t b^{2}\left(6-5 x-x^{2}\right) \). Factor the expression by following steps: - step0: Factor: \(25a^{2}\left(x^{2}+5x-6\right)dtb^{2}\left(6-5x-x^{2}\right)\) - step1: Multiply the terms: \(25a^{2}dtb^{2}\left(x^{2}+5x-6\right)\left(6-5x-x^{2}\right)\) - step2: Multiply the terms: \(25a^{2}dtb^{2}\left(-\left(x^{2}+5x-6\right)^{2}\right)\) - step3: Factor the expression: \(25a^{2}dtb^{2}\left(-\left(x-1\right)^{2}\right)\left(x+6\right)^{2}\) - step4: Simplify the expression: \(-25a^{2}dtb^{2}\left(x-1\right)^{2}\left(x+6\right)^{2}\) Here are the complete factorizations for each of the expressions: ### 2.1 Factor \( x^{4}-49 \) This expression is a difference of squares: \[ x^{4}-49 = (x^{2}-7)(x^{2}+7) \] ### 2.2 Factor \( 20-5x^{2} \) We can factor out the common factor: \[ 20-5x^{2} = 5(4-x^{2}) = 5(2-x)(2+x) \] ### 2.3 Factor \( x^{2}-8x+15 \) This quadratic can be factored as: \[ x^{2}-8x+15 = (x-5)(x-3) \] ### 2.4 Factor \( 24-x^{3}-8^{3} \) This expression can be rewritten as: \[ 24 - x^{3} - 512 = 24 - (x^{3} + 512) \] However, it appears that no result could be found for this input. We can try to factor it using the sum of cubes formula, but it may require additional steps. ### 2.5 Factor \( 25a^{2}(x^{2}+5x-6)dtb^{2}(6-5x-x^{2}) \) This can be factored as: \[ 25a^{2}dtb^{2}(-25)(x-1)^{2}(x+6)^{2} \] Note that the second part \( (6-5x-x^{2}) \) can be rearranged and factored as well. If you need further assistance with the fourth expression or any other part, please let me know!