Final answer:
To factorize the quadratic expression 4c²+10c+4, we find factors of 16 that add up to 10, which are 2 and 8. We then rewrite the original expression as 4c² + 2c + 8c + 4, factor by grouping, and finally end up with (2c + 4)(2c + 1).
Step-by-step explanation:
To factorize the quadratic expression 4c²+10c+4, we need to find two binomials that when multiplied yield the original expression. This requires finding two numbers that both add up to the coefficient of the middle term (10) and multiply to give the product of the first and third coefficients (4 \(\times\) 4 = 16).
Let's find such numbers:
- The factors of 16 that also add up to 10 are 2 and 8 since 2 \(\times\) 8 = 16 and 2 + 8 = 10.
- We then write the middle term (10c) as 2c + 8c.
- Rewriting the expression, we have 4c² + 2c + 8c + 4.
- Now we can factor by grouping: (4c² + 2c) + (8c + 4).
- Factoring out the greatest common factor (GCF) from each group gives us 2c(2c + 1) + 4(2c + 1).
- Finally, we can factor out the common binomial factor to obtain (2c + 4)(2c + 1).
The expression 4c²+10c+4 is fully factorized as (2c + 4)(2c + 1).
In case of confusion, remember to:
- Eliminate terms wherever possible to simplify the algebra.
- Check the answer to see if it is reasonable.