Question

1. factor fully:

a) $x^2 + 7x + 10$
b) $x^2 - 8x + 7$
c) $x^2 + x - 20$
d) $x^2 - 2x - 35$
e) $4x^2 + 8x + 4$

Explanation:

Step1: Factor \(x^2 + 7x + 10\)

We need two numbers that multiply to \(10\) and add to \(7\). The numbers are \(2\) and \(5\). So, \(x^2 + 7x + 10=(x + 2)(x + 5)\).

Step2: Factor \(x^2 - 8x + 7\)

Find two numbers that multiply to \(7\) and add to \(-8\). The numbers are \(-1\) and \(-7\). Thus, \(x^2 - 8x + 7=(x - 1)(x - 7)\).

Step3: Factor \(x^2 + x - 20\)

We need two numbers that multiply to \(-20\) and add to \(1\). The numbers are \(5\) and \(-4\). So, \(x^2 + x - 20=(x + 5)(x - 4)\).

Step4: Factor \(x^2 - 2x - 35\)

Find two numbers that multiply to \(-35\) and add to \(-2\). The numbers are \(-7\) and \(5\). Thus, \(x^2 - 2x - 35=(x - 7)(x + 5)\).

Step5: Factor \(4x^2 + 8x + 4\)

First, factor out the greatest common factor \(4\): \(4(x^2 + 2x + 1)\). Then, factor the quadratic \(x^2 + 2x + 1=(x + 1)^2\). So, \(4x^2 + 8x + 4 = 4(x + 1)^2\).

Answer:

a) \((x + 2)(x + 5)\)
b) \((x - 1)(x - 7)\)
c) \((x + 5)(x - 4)\)
d) \((x - 7)(x + 5)\)
e) \(4(x + 1)^2\) (assuming the last one is part e or a typo in labeling)