Question

1. factor.\

a) $4y^2 - 20y - 56$ \
b) $-3m^2 - 18m - 24$ \
c) $4x^2 + 4x - 48$ \
d) $10x^2 + 80x + 120$ \
e) $-5n^2 + 40n - 35$ \
f) $7c^2 - 35c + 42$

Explanation:

Part a) \(4y^2 - 20y - 56\)

Step 1: Factor out the GCF

The greatest common factor (GCF) of \(4y^2\), \(-20y\), and \(-56\) is \(4\).
\(4y^2 - 20y - 56 = 4(y^2 - 5y - 14)\)

Step 2: Factor the quadratic

We need two numbers that multiply to \(-14\) and add to \(-5\). The numbers are \(-7\) and \(2\).
\(y^2 - 5y - 14=(y - 7)(y + 2)\)

Step 3: Combine

\(4y^2 - 20y - 56 = 4(y - 7)(y + 2)\)

Step 1: Factor out the GCF (negative)

Factor out \(-3\) (to make the leading coefficient positive for easier factoring).
\(-3m^2 - 18m - 24=-3(m^2 + 6m + 8)\)

Step 2: Factor the quadratic

We need two numbers that multiply to \(8\) and add to \(6\). The numbers are \(2\) and \(4\).
\(m^2 + 6m + 8=(m + 2)(m + 4)\)

Step 3: Combine

\(-3m^2 - 18m - 24=-3(m + 2)(m + 4)\)

Step 1: Factor out the GCF

The GCF of \(4x^2\), \(4x\), and \(-48\) is \(4\).
\(4x^2 + 4x - 48 = 4(x^2 + x - 12)\)

Step 2: Factor the quadratic

We need two numbers that multiply to \(-12\) and add to \(1\). The numbers are \(4\) and \(-3\).
\(x^2 + x - 12=(x + 4)(x - 3)\)

Step 3: Combine

\(4x^2 + 4x - 48 = 4(x + 4)(x - 3)\)

Answer:

\(4(y - 7)(y + 2)\)

Part b) \(-3m^2 - 18m - 24\)