Factorize (x+2)^2 + x^2 +6x + 8
=x^2+4x+4+x^2+6x+8
=2x^2+10x+12
=2(x^2+5+6)
=(x+2)(x+3)
=x=-3,-2
picking up from Brian's third-last line
= 2(x^2+5+6)
should have been
= 2(x^2+5x+6)
= 2(x+2)(x+3)
This is the fully factored form, there was no equation to "solve"
To factorize the expression (x+2)^2 + x^2 + 6x + 8, we can follow these steps:
Step 1: Group the terms together based on their similarities.
(x+2)^2 + x^2 + 6x + 8 can be rewritten as:
((x+2)^2 + x^2) + (6x + 8)
Step 2: Simplify each group.
For the first group, we can expand (x+2)^2 using the formula (a+b)^2 = a^2 + 2ab + b^2:
(x^2 + 4x + 4) + x^2
Step 3: Combine the like terms.
We can now combine the like terms in the expression:
2x^2 + 4x + 4 + 6x + 8
Step 4: Simplify further.
Combine the constant terms and the x terms:
2x^2 + 10x + 12
Step 5: Factorize the quadratic expression.
To factorize the quadratic expression, we need to find two numbers that multiply to give the product of the coefficient of the x^2 term (2) and the constant term (12), and add up to give the coefficient of the x term (10).
The factors of 2 and 12 are:
1 and 2
3 and 4
6 and 2
-1 and -12
-2 and -6
-3 and -4
From these factors, we can see that the numbers 2 and 6 add up to 8 (not 10). Therefore, we need to use the factors -2 and -6.
We can rewrite the expression as:
2x^2 - 2x - 6x + 12
Step 6: Factor by grouping.
Now, we can group the terms in pairs and factor them:
(2x^2 - 2x) + (-6x + 12)
2x(x - 1) - 6(x - 2)
Step 7: Combine the factors.
We can now combine the factors to obtain the final factorization:
2(x - 2)(x - 1)
So, the factorization of the given expression (x+2)^2 + x^2 + 6x + 8 is 2(x - 2)(x - 1).