Certainly! We need to factorize the expression [tex]\( 4a^2 + 20a + 25 \)[/tex]. Let's go through the step-by-step process of factorization.
1. Identify the quadratic expression: [tex]\( 4a^2 + 20a + 25 \)[/tex].
2. Look for a common factor: In this case, there is no common factor for all coefficients.
3. Identify the general form of a perfect square trinomial: A perfect square trinomial has the form [tex]\( (ax + b)^2 \)[/tex] which expands to [tex]\( a^2x^2 + 2abx + b^2 \)[/tex].
4. Compare the given quadratic expression with the general form:
- The given quadratic expression has the form [tex]\( 4a^2 + 20a + 25 \)[/tex]:
- [tex]\( 4a^2 \)[/tex] suggests [tex]\( (2a)^2 \)[/tex].
- [tex]\( 25 \)[/tex] suggests [tex]\( 5^2 \)[/tex].
- [tex]\( 20a \)[/tex] suggests [tex]\( 2 \times 2a \times 5 \)[/tex].
5. Verify the middle term: The middle term [tex]\( 20a \)[/tex] is indeed [tex]\( 2 \times (2a) \times 5 \)[/tex]. This confirms that the given quadratic expression is a perfect square trinomial.
6. Write the factorized form: From the observations, we can see the expression [tex]\(4a^2 + 20a + 25\)[/tex] fits the pattern of [tex]\((2a + 5)^2\)[/tex].
Hence, the factorized form of [tex]\( 4a^2 + 20a + 25 \)[/tex] is:
[tex]\[ (2a + 5)^2 \][/tex]
Therefore, the expression [tex]\( 4a^2 + 20a + 25 \)[/tex] factorizes to [tex]\( (2a + 5)^2 \)[/tex].
