Questions: (2x^2+x-21)/(x^2+2x) * (8x+16)/(x-3)

Solution Steps

To simplify the given expression, we need to factor each polynomial in the numerators and denominators. After factoring, we can cancel out any common factors between the numerators and denominators. This will simplify the expression to its simplest form.

We start with the expression

\[ \frac{2x^2 + x - 21}{x^2 + 2x} \cdot \frac{8x + 16}{x - 3}. \]

Factoring each part, we find:

• The numerator \(2x^2 + x - 21\) factors to \((x - 3)(2x + 7)\).
• The denominator \(x^2 + 2x\) factors to \(x(x + 2)\).
• The numerator \(8x + 16\) factors to \(8(x + 2)\).
• The denominator \(x - 3\) remains as is.

Substituting the factored forms into the expression, we have:

\[ \frac{(x - 3)(2x + 7)}{x(x + 2)} \cdot \frac{8(x + 2)}{x - 3}. \]

Next, we can cancel the common factors in the numerator and denominator:

• The factor \((x - 3)\) cancels with \((x - 3)\).
• The factor \((x + 2)\) cancels with \((x + 2)\).

This simplifies our expression to:

\[ \frac{8(2x + 7)}{x}. \]

Final Answer