Question

1. find the product. if possible, write your answer in factored form.

\\(\frac{x^2 + 3x - 4}{x^2 + 4x + 4} \cdot \frac{2x^2 + 4x}{x^2 - 4x + 3} = \square , \\ x \
eq \square\\)

Explanation:

Step1: Factor each quadratic expression

Factor \(x^2 + 3x - 4\): We need two numbers that multiply to \(-4\) and add to \(3\). Those numbers are \(4\) and \(-1\), so \(x^2 + 3x - 4=(x + 4)(x - 1)\).
Factor \(x^2 + 4x + 4\): This is a perfect square trinomial, so \(x^2 + 4x + 4=(x + 2)^2\).
Factor \(2x^2 + 4x\): Factor out \(2x\), so \(2x^2 + 4x = 2x(x + 2)\).
Factor \(x^2 - 4x + 3\): We need two numbers that multiply to \(3\) and add to \(-4\). Those numbers are \(-1\) and \(-3\), so \(x^2 - 4x + 3=(x - 1)(x - 3)\).

Step2: Substitute the factored forms into the expression

The original expression \(\frac{x^2 + 3x - 4}{x^2 + 4x + 4} \cdot \frac{2x^2 + 4x}{x^2 - 4x + 3}\) becomes:
\(\frac{(x + 4)(x - 1)}{(x + 2)^2} \cdot \frac{2x(x + 2)}{(x - 1)(x - 3)}\)

Step3: Cancel out common factors

Cancel out \((x - 1)\) from the numerator and denominator, and cancel out one \((x + 2)\) from the numerator and denominator:
\(\frac{(x + 4)\cancel{(x - 1)}}{(x + 2)^{\cancel{2}}} \cdot \frac{2x\cancel{(x + 2)}}{\cancel{(x - 1)}(x - 3)}=\frac{2x(x + 4)}{(x + 2)(x - 3)}\)

Step4: Determine the values of \(x\) to exclude

We need to exclude values of \(x\) that make any denominator zero.
For \(x^2 + 4x + 4=(x + 2)^2 = 0\), \(x=-2\).
For \(x^2 - 4x + 3=(x - 1)(x - 3)=0\), \(x = 1\) or \(x = 3\).

Answer:

The product is \(\frac{2x(x + 4)}{(x + 2)(x - 3)}\), and \(x
eq -2,1,3\) (we can list the excluded values as \(-2, 1, 3\); if we need to put one in the box for \(x
eq\), we can note the key excluded values, but typically the main ones from the denominators' roots are \(-2, 1, 3\); however, based on the problem's box, maybe the most critical or a common one, but the expression simplifies to \(\frac{2x(x + 4)}{(x + 2)(x - 3)}\) and \(x
eq -2,1,3\)).

(If we need to fill the first box with the factored form and the second with the excluded \(x\) values, the first box is \(\frac{2x(x + 4)}{(x + 2)(x - 3)}\) and the second box can have values like \(-2, 1, 3\) (but since it's a single box, maybe the most relevant, but mathematically the excluded values are \(x
eq -2,1,3\)).)