Final answer:
The student's question involves factoring algebraic expressions in mathematics. The quadratic expression cannot be factorised further, while the second expression is a difference of squares and the third can be factored by taking out the greatest common factor.
Step-by-step explanation:
The subject of the question is factoring algebraic expressions, which is a part of high school mathematics. Factoring is the process of breaking down an expression into its simplest components or 'factors' that, when multiplied together, give you the original expression. Below are the factorisations for each expression given:
- For the quadratic expression (3x^2 + 4x - 7), we need to find two numbers that multiply to give -21 (3 * -7) and add to give 4. Unfortunately, since these two numbers do not exist, the expression is already in its simplest form and cannot be factorised further using integer factors.
- (36y^2 - 4x^2) is a difference of squares and can be factored into (6y + 2x)(6y - 2x).
- For the expression (8p^3 + 16p^4t^3 + 24p^5t), first, factor out the greatest common factor, which is 8p^3, resulting in 8p^3(1 + 2pt^3 + 3p^2t).