Questions: Completely factor 4s^4+8s^3-60s^2 4s^2(s+5)(s-3) 4s^2(s-15)(s+4) 4s^2(s-5)(s+3) This polynomial does not factor.

Transcript text: Completely factor $4 s^{4}+8 s^{3}-60 s^{2}$ $4 s^{2}(s+5)(s-3)$ $4 s^{2}(s-15)(s+4)$ $4 s^{2}(s-5)(s+3)$ This polynomial does not factor.

Solution

Solution Steps

To factor the polynomial $4s^4 + 8s^3 - 60s^2$, first factor out the greatest common factor (GCF) from all terms. Then, factor the resulting polynomial further if possible by looking for patterns or using techniques such as grouping or the quadratic formula.

Step 1: Factor Out the GCF

The polynomial $4s^4 + 8s^3 - 60s^2$ can be simplified by factoring out the greatest common factor (GCF), which is $4s^2$. This gives us: \[ 4s^2(s^2 + 2s - 15) \]

Step 2: Factor the Quadratic

Next, we need to factor the quadratic expression $s^2 + 2s - 15$. We look for two numbers that multiply to $-15$ and add to $2$. The numbers $5$ and $-3$ satisfy these conditions. Thus, we can factor the quadratic as: \[ s^2 + 2s - 15 = (s - 3)(s + 5) \]

Step 3: Combine the Factors

Now, substituting back into our expression, we have: \[ 4s^2(s - 3)(s + 5) \]