Answer:
Step-by-step explanation:
PART A: No, the trinomial 4x^3 + 28x + 49 is not a perfect square trinomial. To be a perfect square trinomial, the first and last terms must be perfect squares and the middle term must be twice the product of the square roots of the first and last terms. In this case, the first and last terms are not perfect squares, and the middle term is not twice the product of the square roots of the first and last terms.
PART B: To factor the expression 6x^2 + 5x - 4, we need to find two binomials whose product is equal to this expression.
First, we can break down the coefficient of x into two numbers whose product is 6 * (-4) = -24 and whose sum is 5. These numbers are 8 and -3.
Next, we use these two numbers to split the middle term 5x into two terms:
6x^2 + 8x - 3x - 4
We can now factor the expression by grouping:
(6x^2 + 8x) - (3x + 4)
2x(3x + 4) - 1(3x + 4)
(2x - 1)(3x + 4)
Therefore, the area of the rectangle can be factored as (2x - 1)(3x + 4).
To determine the dimensions of the rectangle, we set each factor equal to one of the dimensions and solve for the other dimension:
2x - 1 = length
3x + 4 = width
Alternatively, we could switch the factors and get the same answer:
2x - 1 = width
3x + 4 = length
Either way, these are the dimensions of the rectangle, expressed in terms of x.